simplify rational or radical expressions with our free step-by-step math calculator. Let's begin! . Consider the product [latex]{x}^{3}\cdot {x}^{4}[/latex]. While the "Fractional Exponents" calculator and "Solve for Exponents" calculator, assist those with a more advanced understanding of exponents. Products of exponential expressions with the same base can be simplified by adding exponents. Solving equations mean finding the value of the unknown variable given. Determine mathematic problems Determining mathematical problems can be difficult, but with practice it can become easier. Simplifying Expressions This section will provide several examples of how to simplify expressions with exponents including at least one problem about each property given above. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. . solving rational equations on ti 89. exponents + adding, subtracting, multiplying, dividing. In a similar way to the product rule, we can simplify an expression such as \displaystyle \frac { {y}^ {m}} { {y}^ {n}} ynym, where \displaystyle m>n m > n. Both terms have the same base, x, but they are raised to different exponents. To simplify expressions, we combine all the like terms and solve all the given brackets, if any, and then in the simplified expression, we will be only left with unlike terms that cannot be reduced further. Consider the expression [latex]{\left({x}^{2}\right)}^{3}[/latex]. simplify rational or radical expressions with our free step-by-step math First Law of Exponents If a and b are positive integers and x is a real number. Simplifying radical expressions calculator Free radical equation calculator - solve radical equations step-by-step. This calculator will solve your problems. Know the order of operations. Being able to simplify expressions not only makes solving equations easier, but it also helps to improve your understanding of math concepts and how they apply to real-world problems. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number. Remember, we're simplifying using positive exponents, so we need to change x^-4. Mathematics is the study of numbers and their relationships. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers. What our customers say Math app provides students with the tools they need to understand and solve their math problems, this app has been very helpful. Exponents In this expression, 6x and -3x are like terms, and -x2 and x2 are like terms. It does not intend to find the value of an unknown quantity. If you're having problems memorizing these properties, I suggest using flash cards. Go! Free Worksheets Order Operations, practice simplifying expression with exponents problems, online dirac laplace calculator. Otherwise, the difference [latex]m-n[/latex] could be zero or negative. In these cases, further simplification is not possible. There's one exponent in this equation: 42, or four to the second power. simplify rational or radical expressions with our free step-by-step math calculator. Step 2: Click the blue arrow to submit and see the result! Simplify the expression using the properties of exponents calculator - Solve equations with PEMDAS order of operations showing the work. An error occurred while processing this operation. The algebra section allows you to expand, factor or simplify virtually any expression you choose. Get detailed solutions to your math problems with our Combining like terms step-by-step calculator. This gives us y ^8-3. In a similar way to the product rule, we can simplify an expression such as [latex]\frac{{y}^{m}}{{y}^{n}}[/latex], where [latex]m>n[/latex]. How to use the calculator to simplify algebraic expressions? Welcome to our step-by-step math solver! Our community of experts can help you with any question you have. Step 2: Click "Simplify" to get a simplified version of the entered expression. Simplifying Exponents. Free simplify calculator - simplify algebraic expressions step-by-step. Exponent Base & Type | What is a Positive Exponent? Example: 2x-1=y,2y+3=x New Example Keyboard Solve e i s c t l L Search Engine users found our website today by entering these keyword phrases : For any real numbers [latex]a[/latex] and [latex]b[/latex], where [latex]b\neq0[/latex], and any integer [latex]n[/latex], the power of a quotient rule of exponents states that. To see how this is done, let us begin with an example. It requires one to be familiar with the concepts of arithmetic operations on algebraic expressions, fractions, and exponents. The simplify calculator will then show you the steps to, The power rule applies to exponents. Multiplying straight across, our final answer is 1/3x^2. When simplifying expressions with exponents, rather than trying to work robotically from the rules, instead think about what the exponents mean. Simplifying radical expressions (addition) Google Classroom About Transcript A worked example of simplifying an expression that is a sum of several radicals. For example, can we simplify [latex]\frac{{h}^{3}}{{h}^{5}}[/latex]? Using the Power Rule to Simplify Expressions With Exponents. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. 2 42 + 18 / 6 - 30. Next, we separate them into multiplication: 16/8 times p/p^3 times q^2 / q^4 times r^9. Find the total cost of buying pencils by both of them. In this blog post, we will be discussing How to simplify expressions with exponents calculator. Simplify each of the following quotients as much as possible using the power of a quotient rule. Now let's look at a couple of examples! In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. Suppose you want the value y x. Solution: This expression only has the denominator with a negative exponent. Step 2: Now click the button "Solve" to get the result. Since we have x^3 divided by x^7, we subtract their exponents. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Finally, our last step - multiplying the fractions straight across. Simplify Calculator Simplify algebraic expressions step-by-step full pad Examples Related Symbolab blog posts Just like numbers have factors (23=6), expressions have factors ( ` . simplify rational or radical expressions with our free step-by-step math First Law of Exponents If a and b are positive integers and x is a real number Deal with math question Math is a subject that often confuses students. When they are, the basic rules of exponents and exponential notation apply when writing and simplifying algebraic expressions that contain exponents. Math is a subject that often confuses students. To simplify algebraic expressions, follow the steps given below: Step 1: Solve parentheses by adding/subtracting like terms inside and by multiplying the terms inside the brackets with the factor written outside. . The calculator displays 1.304596316E13. A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction barfrom numerator to denominator or vice versa. This is true for any nonzero real number, or any variable representing a nonzero real number. Simplifying radical expressions calculator This calculator simplifies expressions that contain radicals. While simplifying expressions with fractions, we have to make sure that the fractions should be in the simplest form and only unlike terms should be present in the simplified expression. By using these properties, you can simplify complex expressions containing logarithms. It works with polynomials with more than one variable as well. There are rules in algebra for simplifying exponents with different and same bases that we can use. Look at the image given below showing another simplifying expression example. We distribute the exponent to everything in the parenthesis. If you're looking for help with your homework, our team of experts have you covered. Example 1: Find the simplified form of the expression formed by the following statement: "Addition of k and 8 multiplied by the subtraction of k from 16". Free simplify calculator - simplify algebraic expressions step-by-step. This can help you to develop a deeper understanding of math and how it applies to the real world, which can be useful in a variety of fields such as science, engineering, and finance. ( ) Simplify an expression or cancel an expression means reduce it by grouping terms. But there is support available in the form of. This typically involves combining like terms (terms with the same variables and exponents), removing unnecessary constants or terms, and rearranging the expression in a more convenient form. Introduction Exponents can be attached to variables as well as numbers. To use the Simplify Calculator, simply enter your expression into the input field and press the Calculate button. Free simplify calculator - simplify algebraic expressions step-by-step. Check these interesting articles related to the concept of simplifying expressions in math. Contains a great and useful calculator, this is one of the best apps relating to education no other app compares with this app it helped me to understand my work better it even shows how it was worked out I recommend to 7 of my friends and they are happy about this app. Explore the use of several properties used to simplify expressions with exponents, including the product of powers, power to a power, quotient of powers, power of a product, and the zero property. We find that [latex]{2}^{3}[/latex] is 8, [latex]{2}^{4}[/latex] is 16, and [latex]{2}^{7}[/latex] is 128. This Simplify exponents expressions calculator supplies step-by-step instructions for solving all math troubles. The calculator allows with this computer algebra function of reducing an algebraic expression. Simplify The calculator will simplify the equation step-by-step, and display the result. Mathematicians, scientists, and economists commonly encounter very large and very small numbers. According to the order of operations, next we'll simplify any exponents. Simplifying Expressions with Distributive Property, Addition and subtraction of algebraic expressions. Simplify Another useful result occurs if we relax the condition that [latex]m>n[/latex] in the quotient rule even further. We strive to deliver products of the highest quality to our customers. | 10 Therefore, 4ps - 2s - 3(ps +1) - 2s = ps - 4s - 3. The calculator will show you all the steps and easy-to-understand explanations of how to simplify polynomials. Being a virtual student, it's been able to help study and understand and breakdown concepts that I was not previously aware of. This is in simplified form using positive exponents. When simplifying math expressions, you can't simply proceed from left to right, multiplying, adding, subtracting, and so on as you go. Homework is a necessary part of school that helps students review and practice what they have learned in class. Solve - Properties of rational exponents calculator. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents. Simplify We're almost done: 2 times p^(1-3) is -2, times q^(2-4), which is q^(-2) times r^9. By simplifying the expression, you can eliminate unnecessary terms and constants, making it easier to focus on the important parts of the equation and work through it step by step. Step 1: Enter the expression you want to simplify into the editor. The expression inside the parentheses is multiplied twice because it has an exponent of 2. succeed. Work on the task that is enjoyable to you Mathematics is the study of numbers, shapes, and patterns. - Definition & Examples, Expressing Relationships as Algebraic Expressions, Practice Simplifying Algebraic Expressions, Expanding & Simplifying Algebraic Expressions, Translating an Addition Statement into an Algebraic Expression, Roots and Powers of Algebraic Expressions, Translating a Division Statement into an Algebraic Expression, Taking the Derivative of arcsin: How-To & Tutorial, Working Scholars Bringing Tuition-Free College to the Community. Simplify each of the following products as much as possible using the power of a product rule. This is the product rule of exponents. But it may not be obvious how common such figures are in everyday life. The goal of simplification is to make the expression easier to work with and understand, while still representing the same value. Let's assume we are now not limited to whole numbers. For instance, consider [latex]{\left(pq\right)}^{3}[/latex]. She holds a master's degree in Learning and Technology. When we use rational exponents, we can apply the properties of exponents to simplify expressions. Then it must be that ( 8 1 3) 3 = 8 3. Whether you are a student working on math assignments or a professional dealing with equations as part of your job, learning to simplify expressions is a valuable investment in your mathematical education and career. To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. simplify rational or radical expressions with our free step-by-step math calculator. Examples Simplify Simplify Simplify Type ^ for exponents like x^2 for x squared. Here's an example: Enter 10, press the exponent key, then press 5 and enter. Use properties of rational exponents to simplify the expression calculator - Practice your math skills and learn step by step with our math solver. Typing Exponents. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction. One of the main benefits of simplifying expressions is that it can save you time and effort. First, we open the brackets, if any. To simplify your expression using the Simplify Calculator, type in your expression like 2 (5x+4)-3x. In this case, you multiply the exponents. So, y/2 4x/1 = (y 4x)/2 = 4xy/2 = 2xy. Therefore, - k2 + 8k + 128 is the simplified form of the given expression. Simplify Calculator Exponents are supported on variables using the ^ (caret) symbol. This appears later in more advanced courses, but for now, we will consider the value to be undefined. Perform the division by canceling common factors. Keep in mind that simplification is not always possible, and sometimes an expression may be already in its simplest form. Remove unnecessary terms: If a term has a coefficient of 0, it can be removed from the expression since it has no effect on the value. Expressions can be rewritten using exponents to be simplified visually and mathematically. To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. The general rule to simplify expressions is PEMDAS - stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Simplify the math operation ie., on multiplying the two large exponents, we will get the final output. By following these steps, you should be able to simplify most algebraic expressions. This video looks at how to work with expressions that have rational exponents (fractions in the exponent). The calculator will show you each step with easy-to-understand explanations . Simplify Expressions With Zero Exponents. Next, x^2 divided by x^4 is x^(2-4). With a negative exponent, this causes the expression to reciprocate and change exponent to positive, so start with 1/ (4096)^ (5/6) = 1/4^5 = 1/1024. To simplify algebraic expressions, follow the steps given below: Let us take an example for a better understanding. Overall, simplifying algebraic expressions is an important skill that can help you to save time, improve your understanding of math, and develop your problem-solving skills. Simplify radical,rational expression with Step. Multi-Step Equations with Fractions & Decimals | Solving Equations with Fractions. The equations section lets you solve an equation or system of equations. Step 1: Enter the algebraic expression in the corresponding input box. Write answers with positive exponents. a. n times. Get math help online by chatting with a tutor or watching a video lesson. How to Use the Negative The exponent calculator simplifies the given exponential expression using the laws of exponents. There are a lot of letters and numbers here, but don't let them trick you. 5/15 reduces to 1/3. This tool is designed to take the frustration out of algebra by helping you to simplify and reduce your expressions to their simplest form. But we know also ( 8 3) 3 = 8. Created by Sal Khan and Monterey Institute for Technology and Education. Exponents & Radicals Calculator. The exponent calculator simplifies the given exponential expression using the laws of exponents. On the top, I have x^3y^8. Simplify is the same as reducing to lowest terms when we talk about fractions. The rules for exponents may be combined to simplify expressions. For any real number [latex]a[/latex] and positive integers [latex]m[/latex] and [latex]n[/latex], the power rule of exponents states that. For example, to express x2, enter x^2. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, Factoring with FOIL, Graphing Parabolas and Solving Quadratics. BYJU'S online negative exponents calculator tool makes the calculation faster, and it displays the result in a fraction of seconds. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. The simplified expression will only have unlike terms connected by addition/subtraction operators that cannot be simplified further. Each piece of the equation fits together to create a complete picture. If you're looking for a tutor who can help you with any subject, look no further than Instant Expert Tutoring. Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057.. Some of the rules for simplifying expressions are listed below: To simplify expressions with exponents is done by applying the rules of exponents on the terms. How to Define a Zero and Negative Exponent, How to Simplify Expressions with Exponents, Simplifying Expressions with Rational Exponents, How to Graph Cubics, Quartics, Quintics and Beyond, How to Add, Subtract and Multiply Polynomials, How to Divide Polynomials with Long Division, How to Use Synthetic Division to Divide Polynomials, Remainder Theorem & Factor Theorem: Definition & Examples, Dividing Polynomials with Long and Synthetic Division: Practice Problems, Practice Problem Set for Exponents and Polynomials, Introduction to Statistics: Tutoring Solution, Study.com ACT® Test Prep: Help and Review, Prentice Hall Algebra 2: Online Textbook Help, College Preparatory Mathematics: Help and Review, McDougal Littell Pre-Algebra: Online Textbook Help, High School Algebra II: Homeschool Curriculum, How to Write a Numerical Expression? If there is a positive sign outside the bracket, then remove the bracket and write all the terms retaining their original signs. Simplify 2n(n2+3n+4) Solution: By using the rules of simplifying expressions, 4ps - 2s - 3(ps +1) - 2s can be simplified as. And if there is a number or variable written just outside the bracket, then multiply it with all the terms inside using the distributive property. To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents. 2 (24 - 20)2 + 18 / 6 - 30. The cost of all 5 pencils = $5x. In math, simplifying expressions is a way to write an expression in its lowest form by combining all like terms together. We have shown that the exponential expression [latex]{a}^{n}[/latex] is defined when [latex]n[/latex] is a natural number, 0, or the negative of a natural number. Its like a teacher waved a magic wand and did the work for me. Create your account, 13 chapters | You can use math to determine all sorts of things, like how much money you'll need to save for a rainy day. There are several steps you can follow to simplify an algebraic expression: Combine like terms: The first step in simplifying an expression is to look for terms with the same variables and exponents and combine them using the appropriate operations. Addition & Subtraction of Rational Exponents, Adding & Subtracting Rational Expressions | Formula & Examples, Algebra Word Problems Help & Answers | How to Solve Word Problems, Multiplying Radical Expressions | Variables, Square Roots & Binomials, Simplifying Algebraic Expressions | Overview, Formulas & Examples. Simplifying dividing algebraic expressions, solve 3x3 systems of linear equations with TI-84 calculator, solving parabola functions, Easiest way to Factor a third-degree polynomial. When you multiply monomial expressions, add the exponents of like bases. The best oart of it is, that it shows the steps to the solutions, not like a regular calculator, i suck at math and I'm a senior in high school about to go in junior year and this helps a lot considering teachers at my school aren't much help anyways, I'm just glad this app exists. And, Victoria bought 6 pencils each for $x, so the cost of 6 pencils = $6x. This section will provide several examples of how to simplify expressions with exponents including at least one problem about each property given above. Sort by: Top Voted Questions Tips & Thanks Related Symbolab blog posts . . 638+ Math Specialists 4.8/5 Quality score 85636+ Student Reviews Get Homework Help As a college student who struggles with algebra like, bUT SOMETIMES THERE ARE SOME PROBLEMS. If we equate the two answers, the result is [latex]{t}^{0}=1[/latex]. Math problems can be determined by using a variety of methods. The procedure to use the negative exponents calculator is as follows: Step 1: Enter the base and exponent value in the respective input field. Need help? [latex]\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1[/latex], If we were to simplify the original expression using the quotient rule, we would have. For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], such that [latex]m>n[/latex], the quotient rule of exponents states that. Here, there are two parentheses both having two unlike terms. Step 2: Click the blue arrow to submit. Estimating Square Roots | How Do You Find the Square Root of a Number? There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. Our first expression has x^3y^8 / y^3x^7. Do not simplify further. Answer Comment ( 3 votes) Upvote Downvote Flag more The exponent of the answer is the product of the exponents: [latex]{\left({x}^{2}\right)}^{3}={x}^{2\cdot 3}={x}^{6}[/latex]. Exponent rules can be used to simplify terms with exponents. The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. Simplifying expressions with exponents In the term , is the base and is the exponent. Exponents are supported on variables using the ^ (caret) symbol. Let us take another example of simplifying 4(2a + 3a + 4) + 6b using the distributive property. . Example 2: Simplify the expression: 4ps - 2s - 3(ps +1) - 2s . Therefore, we move the denominator to the numerator with a positive exponent : Now, we only have positive exponents and we can apply the product of exponents rule to simplify: In just five seconds, you can get the answer to any question you have. Look at the above examples, and see whether and how we have used this property for the simplification of expressions. We start at the beginning. When using the product rule, different terms with the same bases are raised to exponents. It helped me pass my exam and the test questions are very similar to the practice quizzes on Study.com. Solution: Given, Daniel bought 5 pencils each for $x. This calculator will solve your problems. If you want to simplify normal exponents expression without performing any addition, subtraction, multiplication, etc. Understanding of terms with exponents and exponent rules. Check out all of our online calculators here! expression calculator synthetic division calculator program multiply expressions with fractional exponents. a n = a a . [latex]\begin{array}\text{ }x^{3}\cdot x^{4}\hfill&=\stackrel{\text{3 factors } \text{ 4 factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ \hfill& =\stackrel{7 \text{ factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ \hfill& =x^{7}\end{array}[/latex], [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex], [latex]{2}^{3}\cdot {2}^{4}={2}^{3+4}={2}^{7}[/latex]. Simplify expressions with positive exponents calculator - This Simplify expressions with positive exponents calculator helps to fast and easily solve any math. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Let's keep simplifying. Looking for support from expert professors? If so, then you will love the Simplify Calculator. 118 lessons Notice we get the same result by adding the three exponents in one step. Various arithmetic operations like addition, subtraction, multiplication, and division can be applied to simplify . An error occurred trying to load this video. Let's try the best Simplify expressions . Mathematics is a way of dealing with tasks that involves numbers and equations. A really good app really I always used it for school (for a good benefit of course) and it really helps me understand math, although the camera sometimes takes a while to focus, I don't see this as a big issue. Let us learn more about simplifying expressions in this article. This gives us 1/3 times 1/x^2 times 1. By using the product rule of exponents, it can be written as 2ab + 4b3 - 8ab, which is equal to 4b3 - 6ab. Basic knowledge of algebraic expressions is required. Our final, simplified answer is y^5 / x^4. What would happen if [latex]m=n[/latex]? The product [latex]8\cdot 16[/latex] equals 128, so the relationship is true. Use the distributive property to multiply any two polynomials. If the numerator and denominator of the resulting fraction are both divisible by the same number, simplify the fraction by dividing both by that number. Simplify exponential expressions calculator Try the Free Math Solver or Scroll down to Tutorials! Simplify expressions with negative exponents calculator - Apps can be a great way to help learners with their math. This website helped me pass! There are many ways to improve your writing skills, but one of the most effective is to practice regularly. [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}[/latex], [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[/latex], [latex]\begin{array}\text{ }\frac{y^{9}}{y^{5}}\hfill&=\frac{y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y}{y\cdot y\cdot y\cdot y\cdot y} \\ \hfill&=\frac{\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot y\cdot y\cdot y\cdot y}{\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}} \\ \hfill& =\frac{y\cdot y\cdot y\cdot y}{1} \\ \hfill& =y^{4}\end{array}[/latex], [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], [latex]\frac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[/latex]. [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}=\frac{{f}^{14}}{{e}^{14}}[/latex], [latex]\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex], [latex]\begin{array}{ccc}\hfill {\left({e}^{-2}{f}^{2}\right)}^{7}& =& {\left(\frac{{f}^{2}}{{e}^{2}}\right)}^{7}\hfill \\ & =& \frac{{\left({f}^{2}\right)}^{7}}{{\left({e}^{2}\right)}^{7}}\hfill \\ & =& \frac{{f}^{2\cdot 7}}{{e}^{2\cdot 7}}\hfill \\ & =& \frac{{f}^{14}}{{e}^{14}}\hfill \end{array}[/latex], [latex]{\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}[/latex], CC licensed content, Specific attribution, http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2, http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface, [latex]\left(3a\right)^{7}\cdot\left(3a\right)^{10} [/latex], [latex]\left(\left(3a\right)^{7}\right)^{10} [/latex], [latex]\left(3a\right)^{7\cdot10} [/latex], [latex]{\left(a\cdot b\right)}^{n}={a}^{n}\cdot {b}^{n}[/latex], [latex]\left(-3\right)^{5}\cdot \left(-3\right)[/latex], [latex]{x}^{2}\cdot {x}^{5}\cdot {x}^{3}[/latex], [latex]{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}[/latex], [latex]{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}[/latex], [latex]{\left(\frac{2}{y}\right)}^{4}\cdot \left(\frac{2}{y}\right)[/latex], [latex]{t}^{3}\cdot {t}^{6}\cdot {t}^{5}[/latex], [latex]{\left(\frac{2}{y}\right)}^{5}[/latex], [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex], [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex], [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}={\left(-2\right)}^{14 - 9}={\left(-2\right)}^{5}[/latex], [latex]\frac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[/latex], [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}={\left(z\sqrt{2}\right)}^{5 - 1}={\left(z\sqrt{2}\right)}^{4}[/latex], [latex]\frac{{\left(-3\right)}^{6}}{-3}[/latex], [latex]\frac{{\left(e{f}^{2}\right)}^{5}}{{\left(e{f}^{2}\right)}^{3}}[/latex], [latex]{\left(e{f}^{2}\right)}^{2}[/latex], [latex]{\left({x}^{2}\right)}^{7}[/latex], [latex]{\left({\left(2t\right)}^{5}\right)}^{3}[/latex], [latex]{\left({\left(-3\right)}^{5}\right)}^{11}[/latex], [latex]{\left({x}^{2}\right)}^{7}={x}^{2\cdot 7}={x}^{14}[/latex], [latex]{\left({\left(2t\right)}^{5}\right)}^{3}={\left(2t\right)}^{5\cdot 3}={\left(2t\right)}^{15}[/latex], [latex]{\left({\left(-3\right)}^{5}\right)}^{11}={\left(-3\right)}^{5\cdot 11}={\left(-3\right)}^{55}[/latex], [latex]{\left({\left(3y\right)}^{8}\right)}^{3}[/latex], [latex]{\left({t}^{5}\right)}^{7}[/latex], [latex]{\left({\left(-g\right)}^{4}\right)}^{4}[/latex], [latex]\frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}[/latex], [latex]\frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}[/latex], [latex]\begin{array}\text{ }\frac{c^{3}}{c^{3}} \hfill& =c^{3-3} \\ \hfill& =c^{0} \\ \hfill& =1\end{array}[/latex], [latex]\begin{array}{ccc}\hfill \frac{-3{x}^{5}}{{x}^{5}}& =& -3\cdot \frac{{x}^{5}}{{x}^{5}}\hfill \\ & =& -3\cdot {x}^{5 - 5}\hfill \\ & =& -3\cdot {x}^{0}\hfill \\ & =& -3\cdot 1\hfill \\ & =& -3\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \frac{{\left({j}^{2}k\right)}^{4}}{\left({j}^{2}k\right)\cdot {\left({j}^{2}k\right)}^{3}}& =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{1+3}}\hfill & \text{Use the product rule in the denominator}.\hfill \\ & =& \frac{{\left({j}^{2}k\right)}^{4}}{{\left({j}^{2}k\right)}^{4}}\hfill & \text{Simplify}.\hfill \\ & =& {\left({j}^{2}k\right)}^{4 - 4}\hfill & \text{Use the quotient rule}.\hfill \\ & =& {\left({j}^{2}k\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 1& \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \frac{5{\left(r{s}^{2}\right)}^{2}}{{\left(r{s}^{2}\right)}^{2}}& =& 5{\left(r{s}^{2}\right)}^{2 - 2}\hfill & \text{Use the quotient rule}.\hfill \\ & =& 5{\left(r{s}^{2}\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 5\cdot 1\hfill & \text{Use the zero exponent rule}.\hfill \\ & =& 5\hfill & \text{Simplify}.\hfill \end{array}[/latex], [latex]\frac{{\left(d{e}^{2}\right)}^{11}}{2{\left(d{e}^{2}\right)}^{11}}[/latex], [latex]\frac{{w}^{4}\cdot {w}^{2}}{{w}^{6}}[/latex], [latex]\frac{{t}^{3}\cdot {t}^{4}}{{t}^{2}\cdot {t}^{5}}[/latex], [latex]\frac{{\theta }^{3}}{{\theta }^{10}}[/latex], [latex]\frac{{z}^{2}\cdot z}{{z}^{4}}[/latex], [latex]\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}[/latex], [latex]\frac{{\theta }^{3}}{{\theta }^{10}}={\theta }^{3 - 10}={\theta }^{-7}=\frac{1}{{\theta }^{7}}[/latex], [latex]\frac{{z}^{2}\cdot z}{{z}^{4}}=\frac{{z}^{2+1}}{{z}^{4}}=\frac{{z}^{3}}{{z}^{4}}={z}^{3 - 4}={z}^{-1}=\frac{1}{z}[/latex], [latex]\frac{{\left(-5{t}^{3}\right)}^{4}}{{\left(-5{t}^{3}\right)}^{8}}={\left(-5{t}^{3}\right)}^{4 - 8}={\left(-5{t}^{3}\right)}^{-4}=\frac{1}{{\left(-5{t}^{3}\right)}^{4}}[/latex], [latex]\frac{{\left(-3t\right)}^{2}}{{\left(-3t\right)}^{8}}[/latex], [latex]\frac{{f}^{47}}{{f}^{49}\cdot f}[/latex], [latex]\frac{1}{{\left(-3t\right)}^{6}}[/latex], [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}[/latex], [latex]\frac{-7z}{{\left(-7z\right)}^{5}}[/latex], [latex]{b}^{2}\cdot {b}^{-8}={b}^{2 - 8}={b}^{-6}=\frac{1}{{b}^{6}}[/latex], [latex]{\left(-x\right)}^{5}\cdot {\left(-x\right)}^{-5}={\left(-x\right)}^{5 - 5}={\left(-x\right)}^{0}=1[/latex], [latex]\frac{-7z}{{\left(-7z\right)}^{5}}=\frac{{\left(-7z\right)}^{1}}{{\left(-7z\right)}^{5}}={\left(-7z\right)}^{1 - 5}={\left(-7z\right)}^{-4}=\frac{1}{{\left(-7z\right)}^{4}}[/latex], [latex]\frac{{25}^{12}}{{25}^{13}}[/latex], [latex]{t}^{-5}=\frac{1}{{t}^{5}}[/latex], [latex]{\left(a{b}^{2}\right)}^{3}[/latex], [latex]{\left(-2{w}^{3}\right)}^{3}[/latex], [latex]\frac{1}{{\left(-7z\right)}^{4}}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}[/latex], [latex]{\left(a{b}^{2}\right)}^{3}={\left(a\right)}^{3}\cdot {\left({b}^{2}\right)}^{3}={a}^{1\cdot 3}\cdot {b}^{2\cdot 3}={a}^{3}{b}^{6}[/latex], [latex]2{t}^{15}={\left(2\right)}^{15}\cdot {\left(t\right)}^{15}={2}^{15}{t}^{15}=32,768{t}^{15}[/latex], [latex]{\left(-2{w}^{3}\right)}^{3}={\left(-2\right)}^{3}\cdot {\left({w}^{3}\right)}^{3}=-8\cdot {w}^{3\cdot 3}=-8{w}^{9}[/latex], [latex]\frac{1}{{\left(-7z\right)}^{4}}=\frac{1}{{\left(-7\right)}^{4}\cdot {\left(z\right)}^{4}}=\frac{1}{2,401{z}^{4}}[/latex], [latex]{\left({e}^{-2}{f}^{2}\right)}^{7}={\left({e}^{-2}\right)}^{7}\cdot {\left({f}^{2}\right)}^{7}={e}^{-2\cdot 7}\cdot {f}^{2\cdot 7}={e}^{-14}{f}^{14}=\frac{{f}^{14}}{{e}^{14}}[/latex], [latex]{\left({g}^{2}{h}^{3}\right)}^{5}[/latex], [latex]{\left(-3{y}^{5}\right)}^{3}[/latex], [latex]\frac{1}{{\left({a}^{6}{b}^{7}\right)}^{3}}[/latex], [latex]{\left({r}^{3}{s}^{-2}\right)}^{4}[/latex], [latex]\frac{1}{{a}^{18}{b}^{21}}[/latex], [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}[/latex], [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}[/latex], [latex]{\left(\frac{-1}{{t}^{2}}\right)}^{27}[/latex], [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}[/latex], [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}[/latex], [latex]{\left(\frac{4}{{z}^{11}}\right)}^{3}=\frac{{\left(4\right)}^{3}}{{\left({z}^{11}\right)}^{3}}=\frac{64}{{z}^{11\cdot 3}}=\frac{64}{{z}^{33}}[/latex], [latex]{\left(\frac{p}{{q}^{3}}\right)}^{6}=\frac{{\left(p\right)}^{6}}{{\left({q}^{3}\right)}^{6}}=\frac{{p}^{1\cdot 6}}{{q}^{3\cdot 6}}=\frac{{p}^{6}}{{q}^{18}}[/latex], [latex]{\\left(\frac{-1}{{t}^{2}}\\right)}^{27}=\frac{{\\left(-1\\right)}^{27}}{{\\left({t}^{2}\\right)}^{27}}=\frac{-1}{{t}^{2\cdot 27}}=\frac{-1}{{t}^{54}}=-\frac{1}{{t}^{54}}[/latex], [latex]{\left({j}^{3}{k}^{-2}\right)}^{4}={\left(\frac{{j}^{3}}{{k}^{2}}\right)}^{4}=\frac{{\left({j}^{3}\right)}^{4}}{{\left({k}^{2}\right)}^{4}}=\frac{{j}^{3\cdot 4}}{{k}^{2\cdot 4}}=\frac{{j}^{12}}{{k}^{8}}[/latex], [latex]{\left({m}^{-2}{n}^{-2}\right)}^{3}={\left(\frac{1}{{m}^{2}{n}^{2}}\right)}^{3}=\frac{{\left(1\right)}^{3}}{{\left({m}^{2}{n}^{2}\right)}^{3}}=\frac{1}{{\left({m}^{2}\right)}^{3}{\left({n}^{2}\right)}^{3}}=\frac{1}{{m}^{2\cdot 3}\cdot {n}^{2\cdot 3}}=\frac{1}{{m}^{6}{n}^{6}}[/latex], [latex]{\left(\frac{{b}^{5}}{c}\right)}^{3}[/latex], [latex]{\left(\frac{5}{{u}^{8}}\right)}^{4}[/latex], [latex]{\left(\frac{-1}{{w}^{3}}\right)}^{35}[/latex], [latex]{\left({p}^{-4}{q}^{3}\right)}^{8}[/latex], [latex]{\left({c}^{-5}{d}^{-3}\right)}^{4}[/latex], [latex]\frac{1}{{c}^{20}{d}^{12}}[/latex], [latex]{\left(6{m}^{2}{n}^{-1}\right)}^{3}[/latex], [latex]{17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}[/latex], [latex]{\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}[/latex], [latex]\left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)[/latex], [latex]{\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}[/latex], [latex]\frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}[/latex], [latex]\begin{array}{cccc}\hfill {\left(6{m}^{2}{n}^{-1}\right)}^{3}& =& {\left(6\right)}^{3}{\left({m}^{2}\right)}^{3}{\left({n}^{-1}\right)}^{3}\hfill & \text{The power of a product rule}\hfill \\ & =& {6}^{3}{m}^{2\cdot 3}{n}^{-1\cdot 3}\hfill & \text{The power rule}\hfill \\ & =& \text{ }216{m}^{6}{n}^{-3}\hfill & \text{Simplify}.\hfill \\ & =& \frac{216{m}^{6}}{{n}^{3}}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill {17}^{5}\cdot {17}^{-4}\cdot {17}^{-3}& =& {17}^{5 - 4-3}\hfill & \text{The product rule}\hfill \\ & =& {17}^{-2}\hfill & \text{Simplify}.\hfill \\ & =& \frac{1}{{17}^{2}}\text{ or }\frac{1}{289}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill {\left(\frac{{u}^{-1}v}{{v}^{-1}}\right)}^{2}& =& \frac{{\left({u}^{-1}v\right)}^{2}}{{\left({v}^{-1}\right)}^{2}}\hfill & \text{The power of a quotient rule}\hfill \\ & =& \frac{{u}^{-2}{v}^{2}}{{v}^{-2}}\hfill & \text{The power of a product rule}\hfill \\ & =& {u}^{-2}{v}^{2-\left(-2\right)}& \text{The quotient rule}\hfill \\ & =& {u}^{-2}{v}^{4}\hfill & \text{Simplify}.\hfill \\ & =& \frac{{v}^{4}}{{u}^{2}}\hfill & \text{The negative exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \left(-2{a}^{3}{b}^{-1}\right)\left(5{a}^{-2}{b}^{2}\right)& =& -2\cdot 5\cdot {a}^{3}\cdot {a}^{-2}\cdot {b}^{-1}\cdot {b}^{2}\hfill & \text{Commutative and associative laws of multiplication}\hfill \\ & =& -10\cdot {a}^{3 - 2}\cdot {b}^{-1+2}\hfill & \text{The product rule}\hfill \\ & =& -10ab\hfill & \text{Simplify}.\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill {\left({x}^{2}\sqrt{2}\right)}^{4}{\left({x}^{2}\sqrt{2}\right)}^{-4}& =& {\left({x}^{2}\sqrt{2}\right)}^{4 - 4}\hfill & \text{The product rule}\hfill \\ & =& \text{ }{\left({x}^{2}\sqrt{2}\right)}^{0}\hfill & \text{Simplify}.\hfill \\ & =& 1\hfill & \text{The zero exponent rule}\hfill \end{array}[/latex], [latex]\begin{array}{cccc}\hfill \frac{{\left(3{w}^{2}\right)}^{5}}{{\left(6{w}^{-2}\right)}^{2}}& =& \frac{{\left(3\right)}^{5}\cdot {\left({w}^{2}\right)}^{5}}{{\left(6\right)}^{2}\cdot {\left({w}^{-2}\right)}^{2}}\hfill & \text{The power of a product rule}\hfill \\ & =& \frac{{3}^{5}{w}^{2\cdot 5}}{{6}^{2}{w}^{-2\cdot 2}}\hfill & \text{The power rule}\hfill \\ & =& \frac{243{w}^{10}}{36{w}^{-4}}\hfill & \text{Simplify}.\hfill \\ & =& \frac{27{w}^{10-\left(-4\right)}}{4}\hfill & \text{The quotient rule and reduce fraction}\hfill \\ & =& \frac{27{w}^{14}}{4}\hfill & \text{Simplify}.\hfill \end{array}[/latex], [latex]{\left(2u{v}^{-2}\right)}^{-3}[/latex], [latex]{x}^{8}\cdot {x}^{-12}\cdot x[/latex], [latex]{\left(\frac{{e}^{2}{f}^{-3}}{{f}^{-1}}\right)}^{2}[/latex], [latex]\left(9{r}^{-5}{s}^{3}\right)\left(3{r}^{6}{s}^{-4}\right)[/latex], [latex]{\left(\frac{4}{9}t{w}^{-2}\right)}^{-3}{\left(\frac{4}{9}t{w}^{-2}\right)}^{3}[/latex], [latex]\frac{{\left(2{h}^{2}k\right)}^{4}}{{\left(7{h}^{-1}{k}^{2}\right)}^{2}}[/latex].