With Cuemath, you will learn visually and be surprised by the outcomes. 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 . By using these properties, you can simplify complex expressions containing logarithms. As a college student who struggles with algebra like, bUT SOMETIMES THERE ARE SOME PROBLEMS. 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. 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. 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]. We begin by using the associative and commutative properties of multiplication to regroup the factors. Our first expression has x^3y^8 / y^3x^7. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. By using the distributive property, the given expression can be written as 3/4x + y/2 (4x) + y/2 (7). The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In other words, [latex]{\left(pq\right)}^{3}={p}^{3}\cdot {q}^{3}[/latex]. We strive to deliver products of the highest quality to our customers. A valid expression needs to contain numbers and symbols, Experts will give you an answer in real-time, Calculating prices using discounts worksheet, Finding point slope form with two points calculator, How to solve inequalities with variables in the denominator, Straight line postcode distance calculator, Time and work difficult questions for cat. Next, we separate them into multiplication: 16/8 times p/p^3 times q^2 / q^4 times r^9. 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: Let's begin! Mathematics is a way of dealing with tasks that involves numbers and equations. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. Simplify is the same as reducing to lowest terms when we talk about fractions. Free simplify calculator - simplify algebraic expressions step-by-step. For example, the expression 3x + 2y 4x + 5y can be simplified by combining like terms to get 3x 4x + 2y + 5y = -x + 7y. Work on the task that is enjoyable to you Mathematics is the study of numbers, shapes, and patterns. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. Enter an exponential expression below which you want to simplify. 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 This gives us y^8-3. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. This section will provide several examples of how to simplify expressions with exponents including at least one problem about each property given above. Simplifying algebraic expressions is a fundamental skill that is essential for anyone working with math, whether you are a student or a professional. There's one exponent in this equation: 42, or four to the second power. Suppose you want the value y x. algebra simplify division equations 6th grade Math TEKS chart source code of rational expression calculator algebraic rational expressions simplifying. Therefore, 4ps - 2s - 3(ps +1) - 2s = ps - 4s - 3. Step 1: Enter the algebraic expression in the corresponding input box. If you're having problems memorizing these properties, I suggest using flash cards. Examples Simplify Simplify Simplify On the top, I have x^3y^8. Step 2: Now click the button "Solve" to get the result. Try refreshing the page, or contact customer support. Being a virtual student, it's been able to help study and understand and breakdown concepts that I was not previously aware of. Step 1: Enter the expression you want to simplify into the editor. Simplify (m14n12)2(m2n3)12 We can always check that this is true by simplifying each exponential expression. Before learning about simplifying expressions, let us quickly go through the meaning of expressions in math. . An expression with a negative exponent is defined as a reciprocal. The equations section lets you solve an equation or system of equations. All rights reserved. [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]. Multi-Step Equations with Fractions & Decimals | Solving Equations with Fractions. When simplifying expressions with exponents, rather than trying to work robotically from the rules, instead think about what the exponents mean. . 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. For any nonzero real number [latex]a[/latex], the zero exponent rule of exponents states that. We made the condition that [latex]m>n[/latex] so that the difference [latex]m-n[/latex] would never be zero or negative. calculate equation by Improve your scholarly performance How to simplify expressions with exponents calculator - Simplifies expressions step-by-step and shows the work! Write answers with positive exponents. Now, combining all the terms will result in 6x - x2 - 3x + x2. simplify rational or radical expressions with our free step-by-step math calculator. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. [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]. Know the order of operations. Now consider an example with real numbers. Using b x b y = b x + y Simplify. simplify, solve for, expand, factor, rationalize. I would definitely recommend Study.com to my colleagues. Can we simplify the result? Solutions Graphing Practice; New Geometry; Calculators; Notebook . Math is a subject that often confuses students. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer [latex]n[/latex]. Practice your math skills and learn step by step with our math solver. Simplify an expression or cancel an expression means reduce it by grouping terms. In these cases, further simplification is not possible. Factoring can help to make the expression more compact and easier to work with. By using the distributive property of simplifying expression, it can be simplified as. If there is a positive sign outside the bracket, then remove the bracket and write all the terms retaining their original signs. Check out. Simplify (x-2x-3)4. The calculator works for both numbers and expressions containing variables. Expand each expression, and then rewrite the resulting expression. For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], the product rule of exponents states that. Along with PEMDAS, exponent rules, and the knowledge about operations on expressions also need to be used while simplifying algebraic expressions. Simplifying Expressions Calculator Exponents are supported on variables using the ^ (caret) symbol. Simplify radical,rational expression with Step. There are a lot of letters and numbers here, but don't let them trick you. Example 2: Simplify the expression: 4ps - 2s - 3(ps +1) - 2s . By learning to identify patterns and relationships, and by using the properties of exponents and logarithms to simplify expressions, you can improve your ability to think critically and solve complex problems. With this algebra simplifier, you can : Simplify an algebraic expression. She holds a master's degree in Learning and Technology. Therefore, - k2 + 8k + 128 is the simplified form of the given expression. Math is the study of numbers, shapes, and patterns. It uses advanced algorithms to quickly and accurately combine like terms and eliminate unnecessary factors, giving you a clean, simplified expression that is much easier to work with. Write each of the following products with a single base. The calculator will simplify the equation step-by-step, and display the result. Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner. Simplify x.x2 Typing Exponents. And, Victoria bought 6 pencils each for $x, so the cost of 6 pencils = $6x. The calculator will show you each step with easy-to-understand explanations . When you enter an expression into the calculator, the calculator will simplify the Exponents are supported on variables using the ^ (caret) symbol. 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. Here's the fun part, simplify. If you're looking for help with your homework, our team of experts have you covered. An example of simplifying algebraic expressions is given below: Great learning in high school using simple cues. Click the blue arrow to submit. If we keep separating the terms and following the properties, we'll be fine. Simplify x(6 - x) can be simplified as 6x - x2, and -x(3 - x) can be simplified as -3x + x2. . The exponent calculator simplifies the given exponential expression using the laws of exponents. Consider the expression [latex]{\left({x}^{2}\right)}^{3}[/latex]. 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. 2 42 + 18 / 6 - 30. 9y + 3 4x 2y 3x 5. Exponent Calculator - Simplify Exponential Expression. In this article, we will be focussing more on how to simplify algebraic expressions. Another useful result occurs if we relax the condition that [latex]m>n[/latex] in the quotient rule even further. Splitting the multiplication gives us x^3 / x^7 times y^8 / y^3. Open up brackets, if any. Then it must be that ( 8 1 3) 3 = 8 3. First, we open the brackets, if any. Exponents Let's try the best Simplify expressions . Next, x^2 divided by x^4 is x^(2-4). Solve - Properties of rational exponents calculator. Simplify any resulting mixed numbers. The calculator displays 1.304596316E13. This calculator will allow compute an simplify numeric expressions that involve exponents. We follow the same PEMDAS rule to simplify algebraic expressions as we do for simple arithmetic expressions. 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 ( ` . For an instance, (2/4)x + 3/6y is not the simplified expression, as fractions are not reduced to their lowest form. Completing a task step-by-step can help ensure that it is done correctly and efficiently. 986+ Experts. . Here, there are two parentheses both having two unlike terms. For example, (3x2)(2x) can be simplified as 6x3. Finally, our last step - multiplying the fractions straight across. What would happen if [latex]m=n[/latex]? Question ID 14047, 14058, 14059, 14046, 14051, 14056, 14057.. This calculator will try to simplify a polynomial as much as possible. Expressions refer to mathematical statements having a minimum of two terms containing either numbers, variables, or both connected through an addition/subtraction operator in between. Solve Now How to Simplify Exponents or Powers on the TI In this blog post, we will be discussing How to simplify expressions with exponents calculator. Simplify the expression using the properties of exponents calculator - Solve equations with PEMDAS order of operations showing the work. Here are the basic steps to follow to simplify an algebraic expression: remove parentheses by multiplying factors use exponent rules to remove parentheses in terms with exponents combine like terms by adding coefficients combine the constants Let's work through an example. To simplify your expression using the Simplify Calculator, type in your expression like 2(5x+4)-3x. See the steps to to. Well, 5 is positive, so we don't need to change it. Free Worksheets Order Operations, practice simplifying expression with exponents problems, online dirac laplace calculator. Do not simplify further. Ok. that was just a quick review. 1 comment ( 7 votes) Upvote Downvote Flag more htom 2 years ago well what if something was like 1/2 to the power of 7 how would you solve that? Step 2: Click "Simplify" to get a simplified version of the entered expression. Our final, simplified answer is y^5 / x^4. Simplifying Expressions Calculator is a free online tool that displays the simplification of the given algebraic expression. Simplify, Simplify (a12b)12(ab12) Quick-Start Guide Enter an equation in the box, then click "SIMPLIFY". . The exponent calculator simplifies the given exponential expression using the laws of exponents. Note: exponents must be positive integers, no negatives. Simplify the expression \frac { { { {x}^ {2}}}} { { { {x}^ { {-3}}}}} x3x2. a1 n = na. This gives us y ^8-3. Look at the image given below showing another simplifying expression example. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. Write answers with positive exponents. Expand and simplify polynomials. After this lesson you'll be able to simplify expressions with exponents. Remember, we're simplifying using positive exponents, so we need to change x^-4. So, we will be solving the brackets first by multiplying x to the terms written inside. Perform the division by canceling common factors. Yes. If you want to simplify normal exponents expression without performing any addition, subtraction, multiplication, etc. Here is an example: 2x^2+x(4x+3) Need more problem types? If you're looking for a tutor who can help you with any subject, look no further than Instant Expert Tutoring. Next step - look at each part individually. Sort by: Top Voted Questions Tips & Thanks Example 3: Daniel bought 5 pencils each costing $x, and Victoria bought 6 pencils each costing $x. Return to the quotient rule. But it may not be obvious how common such figures are in everyday life. 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 +. Free Exponents Calculator - Simplify exponential expressions using algebraic rules step-by-step. When you multiply monomial expressions, add the exponents of like bases. This is the product rule of exponents. This is our simplified answer with positive exponents. To unlock this lesson you must be a Study.com Member. Complex numbers involve the quantity known as i , an "imaginary" number with the property i = 1.If you have to simply an expression involving a complex number, it might seem daunting, but it's quite a simple process once you learn the basic rules. . Simplification can also help to improve your understanding of math concepts. Simplify For example, you can combine 3x and 2x by adding them to get 3x + 2x = 5x. Looking for help with your math homework? Created by Sal Khan and Monterey Institute for Technology and Education. Simplify each of the following quotients as much as possible using the power of a quotient rule. [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].
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