MathCelebrity.com's Simplify Radical Expressions Calculator - This calculator provides detailed . For example, can we simplify [latex]\frac{{h}^{3}}{{h}^{5}}[/latex]? For those who need an instant solution, we have the perfect answer. Whether you are a student working on a math assignment or a professional dealing with equations as part of your job, the Simplify Expression Calculator is an essential tool that can save you time and make solving equations much easier. We are asked to simplify using positive exponents: p^(-2) is the same as 1/p^2; q^(-2) is the same 1/q^2. Use this, i was struggling with simplifying but this calculator has everything needed, this app was amazing and the best responses and the best Solutions I would refer this to everyone . Simplify algebraic expressions with exponents. The procedure to use the simplifying expressions calculator is as follows: Step 1: Enter the expression in the respective input field Step 2: Now click the button "Submit" to get the result Step 3: Finally, the simplified expression will be displayed in the new window What is Meant by Simplifying Expressions? 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. Step 1, how do i find my safe credit union account number, how to write a number in expanded form in two ways, simplify expressions with rational exponents calculator. Welcome to our step-by-step math solver! You can improve your academic performance by studying regularly and attending class. Get detailed solutions to your math problems with our Combining like terms step-by-step calculator. Note: exponents must be positive integers . But there is support available in the form of. Simplify Expressions With Zero Exponents. Simplify each expression using the zero exponent rule of exponents. Our first expression has x^3y^8 / y^3x^7. Step 1: Enter the algebraic expression in the corresponding input box. Our community of experts can help you with any question you have. Quality is important in all aspects of life. Know the order of operations. Try refreshing the page, or contact customer support. Let's rewrite this with like terms over each other: 5/15 times x^2 / x^4 times y^9/y^9. We start at the beginning. Exponents are supported on variables using the ^ (caret) symbol. BYJU'S online negative exponents calculator tool makes the calculation faster, and it displays the result in a fraction of seconds. We distribute the exponent to everything in the parenthesis. In this case, you add the exponents. In addition to its practical benefits, simplifying expressions is also a great way to develop your problem-solving skills. This gives us 1/3 times 1/x^2 times 1. [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]. Here's an example: Enter 10, press the exponent key, then press 5 and enter. 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. . In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. For example, lets look at the following example. Quotients of exponential expressions with the same base can be simplified by subtracting exponents. It requires one to be familiar with the concepts of arithmetic operations on algebraic expressions, fractions, and exponents. We're almost done: 2 times p^(1-3) is -2, times q^(2-4), which is q^(-2) times r^9. Simplify radical,rational expression with Step. Multiplying straight across, our final answer is 1/3x^2. If so, then you will love the Simplify Calculator. Choose "Simplify" from the topic selector and click to see the result in our Algebra Calculator! 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. [latex]\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[/latex]. 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. The exponent calculator simplifies the given exponential expression using the laws of exponents. And, Victoria bought 6 pencils each for $x, so the cost of 6 pencils = $6x. Let me show you another one. Simplify Calculator Exponents are supported on variables using the ^ (caret) symbol. Open up brackets, if any. In this equation, you'd start by simplifying the part of the expression in parentheses: 24 - 20. Simplify (m14n12)2(m2n3)12 simplify rational or radical expressions with our free step-by-step math calculator. calculate equation by Improve your scholarly performance For example, you can combine 3x and 2x by adding them to get 3x + 2x = 5x. Get math help online by chatting with a tutor or watching a video lesson. If we equate the two answers, the result is [latex]{t}^{0}=1[/latex]. Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend. In this case, you multiply the exponents. Simplifying Exponents. I highly recommend you use this site! Now, to multiply fractions, we multiply the numerators and the denominators separately. Check out. . The calculator above accepts negative bases, but does not compute imaginary numbers. In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex]. Understanding of terms with exponents and exponent rules. Simplifying algebraic expressions is a fundamental skill that is essential for anyone working with math, whether you are a student or a professional. Use the properties of exponents: If an expression contains exponents, you can use the properties of exponents to simplify it. Step 2: Click "Simplify" to get a simplified version of the entered expression. The calculator will then show you the simplified version of the expression, along with a step-by-step breakdown of the simplification process. Math is a subject that often confuses students. Math problems can be determined by using a variety of methods. For example, to express x2, enter x^2. BYJU'S online simplifying. I would definitely recommend Study.com to my colleagues. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction. Example 2: Simplify the expression: 4ps - 2s - 3(ps +1) - 2s . At first, it may appear that we cannot simplify a product of three factors. Otherwise, the difference [latex]m-n[/latex] could be zero or negative. If we keep separating the terms and following the properties, we'll be fine. Simplify simplify, solve for, expand, factor, rationalize. Our support team is available 24/7 to assist you. When you multiply monomial expressions, add the exponents of like bases. Write each of the following products with a single base. Simplify 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. Simplifying Radical Expressions replace the square root sign ( ) with the letter r. show help examples Preview: Input Expression: Examples: r125 8/r2 (1+2r2)^2 Our first step is to simplify (2p)^3. Math understanding that gets you Simplifying expressions with exponents calculator - Here, we debate how Simplifying expressions with exponents calculator can help students learn Algebra. Mathematicians, scientists, and economists commonly encounter very large and very small numbers. There's one exponent in this equation: 42, or four to the second power. For example, to express x2, enter x^2. When you are working with a simplified expression, it is easier to see the underlying patterns and relationships that govern the equation. Multiplying Exponents | How to Multiply Exponents With Different Bases. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. When we use rational exponents, we can apply the properties of exponents to simplify expressions. Definition 17.4.1: Rational Exponent a1 n. If na is a real number and n 2, then. Free simplify calculator - simplify algebraic expressions step-by-step. Mathematics is a way of dealing with tasks that involves numbers and equations. EXAMPLE 1. 2 2 = 2 2 = 4 Square Root Calculator Calculate real and complex square roots (2nd order roots) of numbers or x. You can use the keyboard to enter exponents, fractions, and parentheses, among others. Consider the example [latex]\frac{{y}^{9}}{{y}^{5}}[/latex]. An error occurred while processing this operation. 5/15 reduces to 1/3. This website uses cookies to ensure you get the best experience on our website. Variables Any lowercase letter may be used as a variable. Exponent Base & Type | What is a Positive Exponent? Before learning about simplifying expressions, let us quickly go through the meaning of expressions in math. For example, the expression 4x + 3y + 6x can be simplified by factoring out the common factor 2x to get x(4 + 6) + 3y = 10x + 3y. An expression with exponent zero is defined as 1. Simplifying mathematical expressions implies rewriting the same algebraic statement compactly with no like terms. Simplify each of the following products as much as possible using the power of a product rule. However, when simplifying expressions containing exponents, don't feel like you must work only with, or straight from, these rules. An expression with a negative exponent is defined as a reciprocal. Simplify Calculator. By breaking down and clarifying the steps in a math equation, students can more easily understand and solve the problem. This implies, 2ab + 4b (b2) - 4b (2a). That means that [latex]{a}^{n}[/latex] is defined for any integer [latex]n[/latex]. Mathway requires javascript and a modern browser. The first step I like to do is put the like terms on top of each other. When simplifying math expressions, you can't simply proceed from left to right, multiplying, adding, subtracting, and so on as you go. copyright 2003-2023 Study.com. For example, 3x + 0y can be simplified to 3x. You can also use the calculator to check your work and ensure that you have correctly simplified your expression. To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. When you are working with complex equations, it can be easy to get lost in the details and lose track of what you are trying to solve. To find the product of powersMultiplication of two or more values in exponential form that have the same base-. Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057.. Write each of the following quotients with a single base. Related Symbolab blog posts But we know also ( 8 3) 3 = 8. [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/[email protected], http://cnx.org/contents/[email protected]: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].
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