how to simplify expressions with exponents calculator

Here is an example: 2x^2+x(4x+3) Need more problem types? For those who need an instant solution, we have the perfect answer. Our first step is to simplify (2p)^3. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Looking for help with your math homework? Please ensure that your password is at least 8 characters and contains each of the following: You'll be able to enter math problems once our session is over. It includes four examples. The rules for exponents may be combined to simplify expressions. I can help you determine the answer to math problems. 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. Explore the use of several properties used to simplify expressions with exponents, including the. The sole exception is the expression [latex]{0}^{0}[/latex]. Simplifying Expressions Calculator is a free online tool that displays the simplification of the given algebraic expression. This step is important when you first begin because you can see exactly what we are doing. If you wish to solve the equation, use the Equation Solving Calculator. We're almost done: 2 times p^(1-3) is -2, times q^(2-4), which is q^(-2) times r^9. In math, simplifying expressions is a way to write an expression in its lowest form by combining all like terms together. Check out our online math support services! In this blog post, we will be discussing How to simplify expressions with exponents calculator. Welcome to our step-by-step math solver! This time we have 5x^2y^9 / 15y^9x^4. The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. For example, 2x (x + y) can be simplified as 2x 2 + 2xy. Step 2: Click the blue arrow to submit. Divide one exponential expression by another with a larger exponent. Need help? In the term , is the base and is the exponent. Using a calculator, we enter [latex]2,048\times 1,536\times 48\times 24\times 3,600[/latex] and press ENTER. One way to think about math equations is to think of them as a puzzle. Simplify Radical Expressions Calculator Solve y x n to simplified radical expressions or an integer including complex solutions Square Calculator x Calculate the squared value of integers, decimals and scientific e notation. On the top, I have x^3y^8. In this case, you multiply the exponents. Use the distributive property to multiply any two polynomials. Simplify expressions with negative exponents calculator - Apps can be a great way to help learners with their math. By breaking down and clarifying the steps in a math equation, students can more easily understand and solve the problem. To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. The calculator will show you all the steps and easy-to-understand explanations of how to simplify polynomials. solving rational equations on ti 89. exponents + adding, subtracting, multiplying, dividing. If you're looking for a tutor who can help you with any subject, look no further than Instant Expert Tutoring. Do not simplify further. Therefore, the total cost of pencils bought by them = $5x + $6x = $11x. She holds a master's degree in Learning and Technology. Check out. An expression with a negative exponent is defined as a reciprocal. Math is the study of numbers, shapes, and patterns. 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. Type ^ for exponents like x^2 for "x squared". Type ^ for exponents like x^2 for x squared. Simplify each of the following products as much as possible using the power of a product rule. Enter an exponential expression below which you want to simplify. So, y/2 4x/1 = (y 4x)/2 = 4xy/2 = 2xy. Keep in mind that simplification is not always possible, and sometimes an expression may be already in its simplest form. 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. Really a helpful situation where you can check answers after u solve a problem, and if your wrong, u can always fix it and learn from mistakes using this app, also thank you for the feature of calculating directly from the paper without typing. Using the Power Rule to Simplify Expressions With Exponents. In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex]. If there is a positive sign outside the bracket, then remove the bracket and write all the terms retaining their original signs. Simplify Simplifying radical expressions calculator This calculator simplifies expressions that contain radicals. In the denominator, I want the xs over each other and the ys over each other, so I write x^7y^3. There's one exponent in this equation: 42, or four to the second power. . Basic knowledge of algebraic expressions is required. Simplify expressions with positive exponents calculator - This Simplify expressions with positive exponents calculator helps to fast and easily solve any math. Simplify each expression and write the answer with positive exponents only. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. I would definitely recommend Study.com to my colleagues. After this lesson you'll be able to simplify expressions with exponents. Used with the function expand, the function simplify can expand and collapse a literal expression. Since we have x^3 divided by x^7, we subtract their exponents. Notice we get the same result by adding the three exponents in one step. Step 2: Now click the button "Solve" to get the result. Determine mathematic problems Determining mathematical problems can be difficult, but with practice it can become easier. Volume & Surface Area of a Sphere | How to Find the Surface Area of a Sphere, System of Equations Word Problems & Explanations | How to Solve System of Equations Word Problems, Negative Signs and Simplifying Algebraic Expressions, SAT Subject Test Mathematics Level 2: Practice and Study Guide, GED Math: Quantitative, Arithmetic & Algebraic Problem Solving, CLEP College Algebra: Study Guide & Test Prep, Holt McDougal Algebra 2: Online Textbook Help, McDougal Littell Algebra 2: Online Textbook Help, Algebra II Curriculum Resource & Lesson Plans, Accuplacer Math: Quantitative Reasoning, Algebra, and Statistics Placement Test Study Guide, OSAT Middle Level/Intermediate Mathematics (CEOE) (125): Practice & Study Guide, Explorations in Core Math - Algebra 2: Online Textbook Help, Intermediate Algebra for College Students, NY Regents Exam - Algebra II: Test Prep & Practice, Create an account to start this course today. Introduction Exponents can be attached to variables as well as numbers. Simplify exponential expressions calculator Try the Free Math Solver or Scroll down to Tutorials! simplify rational or radical expressions with our free step-by-step math calculator. By following these steps, you should be able to simplify most algebraic expressions. Therefore, 3/4x + y/2 (4x + 7) = 3/4x + 2xy + 7y/2. Let us take one more example to understand it. Example of Dividing Monomials When you divide monomial expressions, subtract the exponents of like bases. We distribute the exponent to everything in the parenthesis. 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. Step 1: Enter the expression you want to simplify into the editor. Mathematics is the study of numbers and their relationships. Simplifying expressions with exponents In the term , is the base and is the exponent. Exponents are supported on variables using the ^ (caret) symbol. This is in simplified form using positive exponents. 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. Otherwise, the difference [latex]m-n[/latex] could be zero or negative. a n = a a . When using the product rule, different terms with the same bases are raised to exponents. This is how we can simplify expressions with exponents using the rules of exponents. The exponent of the answer is the product of the exponents: [latex]{\left({x}^{2}\right)}^{3}={x}^{2\cdot 3}={x}^{6}[/latex]. Write answers with positive exponents. BYJU'S online simplifying. 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. If there is a negative sign outside the bracket, then remove the bracket and change the signs of all the terms written inside from + to -, and - to +. When one piece is missing, it can be difficult to see the whole picture. Simplify Calculator. We have 1/3 times x^(2-4), which is -2, times y^(9-9), which is y^0. This same logic can be used for any positive integer exponent n to show that a 1 n = a n. RATIONAL EXPONENT a 1 n When we use rational exponents, we can apply the properties of exponents to simplify expressions. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Get math help online by chatting with a tutor or watching a video lesson. Enrolling in a course lets you earn progress by passing quizzes and exams. Our support team is available 24/7 to assist you. Consider the example [latex]\frac{{y}^{9}}{{y}^{5}}[/latex]. Check out all of our online calculators here! 24 minus 20 is 4. Next, we separate them into multiplication: 16/8 times p/p^3 times q^2 / q^4 times r^9. This tool is designed to take the frustration out of algebra by helping you to simplify and reduce your expressions to their simplest form. This Simplify exponents expressions calculator supplies step-by-step instructions for solving all math troubles. Simplify expressions with positive exponents calculator - Math can be a challenging subject for many learners. You can use the keyboard to enter exponents, fractions, and parentheses, among others. The result is that [latex]{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}[/latex]. Kathryn teaches college math. Simplifying radical expressions calculator Free radical equation calculator - solve radical equations step-by-step. The "Exponents" calculator is great for those with a basic understanding of exponents. The simplify calculator will then show you the steps to, The power rule applies to exponents. [latex]\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex]. So, we will be solving the brackets first by multiplying x to the terms written inside. But we know also ( 8 3) 3 = 8. 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. EXAMPLE 1. 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. All three are unlike terms, so it is the simplified form of the given expression. The mathematical concepts that are important in simplifying algebraic expressions are given below: The rules for simplifying expressions are given below: Follow the steps given below to learn how to simplify expressions: Equations refer to those statements that have an equal to "=" sign between the term(s) written on the left side and the term(s) written on the right side. The simplified expression will only have unlike terms connected by addition/subtraction operators that cannot be simplified further. We start at the beginning. Use the zero exponent and other rules to simplify each expression. In this equation, you'd start by simplifying the part of the expression in parentheses: 24 - 20. It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule. Looking for support from expert professors? The cost of all 5 pencils = $5x. 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 : a1 n = na. - 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. How to Solve Exponents Download Article methods 1 Solving Basic Exponents 2 Adding, Subtracting and Multiplying Exponents 3 Solving Fractional Exponents Other Sections Related Articles References Article Summary Co-authored by David Jia Last Updated: February 27, 2023 Exponents are used when a number is multiplied by itself. System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval . An error occurred while processing this operation. Expand and simplify polynomials. Consider the expression [latex]{\left({x}^{2}\right)}^{3}[/latex]. When you are working with a simplified expression, it is easier to see the underlying patterns and relationships that govern the equation. Simplifying Expressions with Distributive Property, Addition and subtraction of algebraic expressions. You can improve your academic performance by studying regularly and attending class. Simplify the expression: x (6 x) x (3 x). This gives us y^8-3. When simplifying expressions with exponents, rather than trying to work robotically from the rules, instead think about what the exponents mean. Exponents Groups Cheat . To simplify a power of a power, you multiply the exponents, keeping the base the same. Using b x b y = b x + y Simplify More ways to get app Simplify Calculator Since we have y ^8 divided by y ^3, we subtract their exponents. [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].