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Chapter 2: Problem 30

Simplify. \(\frac{k^{5}\left(k^{2}\right)^{3}}{7 m^{10}\left(2 m^{3}\right)^{2}}\)

Short Answer

Expert verified
\(\frac{k^{11}}{28 m^4}\)

Step by step solution

01

Apply the Power of a Power Rule

First, we need to apply the power of a power rule to the given expression: \((a^m)^n = a^{m*n}\). In our case, we have \((k^2)^3\) and \((2m^3)^2\). \[\frac{k^{5}(k^{2})^{3}}{7 m^{10}(2 m^{3})^{2}} =\frac{k^{5}k^{2*3}}{7 m^{10}(2^{2} m^{3*2})}\]
02

Calculate the Exponents

Now we can calculate the exponents: \[\frac{k^{5}k^{6}}{7 m^{10}(4 m^{6})}\]
03

Apply the Product of Powers Rule

Next, we need to apply the product of powers rule to the numerator: \(a^m * a^n = a^{m+n}\). In our case, we have \(k^5 * k^6\). \[\frac{k^{5+6}}{7 m^{10}(4 m^{6})} =\frac{k^{11}}{7 m^{10}(4 m^{6})}\]
04

Distribute the Coefficient in the Denominator

Now we need to distribute the coefficient in the denominator: \[\frac{k^{11}}{28 m^{10} m^{6}}\]
05

Apply the Quotient of Powers Rule

Finally, we need to apply the quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\). In our case, we have \(\frac{m^{10}}{m^6}\). \[\frac{k^{11}}{28 m^{10-6}} = \frac{k^{11}}{28 m^{4}}\] So, the simplified expression is: \(\frac{k^{11}}{28 m^4}\).

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Most popular questions from this chapter

Simplify the expression using the product rule. Leave your answer in exponential form. $$t^{5} \cdot t^{4}$$

Perform the operation as indicated. Write the final answer without an exponent. \(\left(2.3 \times 10^{3}\right)\left(3 \times 10^{2}\right)\)

Simplify the expression using the product rule. Leave your answer in exponential form. $$\left(\frac{7}{10} y^{9}\right)\left(-2 y^{4}\right)\left(3 y^{2}\right)$$

Perform the operation as indicated. Write the final answer without an exponent. \(\left(-1.5 \times 10^{-8}\right)\left(4 \times 10^{6}\right)\)

Simplify using the quotient rule. $$\frac{3 a^{2} b^{-5}}{15 a^{-8} b^{3}}$$
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