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Chapter 10: Problem 65

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{-2}$$

Short Answer

Expert verified
\(k^{14}\)

Step by step solution

01

Apply the Power Rule

When raising a power to another power, multiply the exponents. Therefore, \(\big(k^{-7}\big)^{-2} = k^{-7 \times -2} \)
02

Simplify the Exponent

Multiply the exponents: \(-7 \times -2 = 14\). So, \(k^{-7 \times -2} = k^{14}\)
03

Write with Positive Exponents

The expression \(k^{14}\) already has a positive exponent, so no further changes are needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule for Exponents
When you raise a power to another power, you use the Power Rule for exponents. This rule states that you simply multiply the exponents. For example, if you have \(\big(a^m\big)^n\), it's the same as \(a^{m \cdot n}\). This simplifies expressions where exponents are involved.

In the given exercise, we applied the Power Rule to \(\big(k^{-7}\big)^{-2}\). By multiplying the exponents, we get \(k^{-7 \cdot -2} \). This is a crucial step for simplifying expressions efficiently.
Negative Exponents
Negative exponents might look tricky, but they follow simple rules. A negative exponent means that you take the reciprocal of the base. For instance, \(a^{-n} = \frac{1}{a^n}\). One main thing to remember is never to leave negative exponents in your final answer if the problem states to write with positive exponents only.

In our example, we had \(k^{-7}\) raised to \(-2\). After applying the Power Rule, the negative exponents canceled out because \((-7 \cdot -2) = 14\), which converted to a positive exponent \(k^{14}\).
Positive Exponents
Positive exponents indicate that you multiply the base by itself a certain number of times. For example, \(a^3\) means you multiply \(a\) by itself three times: \(a \cdot a \cdot a\).

To wrap up our exercise, converting \(k\) to a term with a positive exponent involved simplifying \((-7 \cdot -2 = 14)\). Our final result was \(k^{14}\), which is already in an ideal form with positive exponents. No need for further changes, and you have your simplified solution!

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Most popular questions from this chapter

Simplify the expression. $$\left(-10 w^{2} z^{3}\right)^{3}$$

Fill in the missing polynomial. $$(x+3)(\quad)=x^{2}+8 x+15$$

Subtract the polynomials. $$\begin{array}{r}15 a^{3}-2 a^{2}+4 a \\\\-\left(2 a^{3}+6 a^{2}-12 a\right) \\\\\hline\end{array}$$

Simplify. Write the answers with positive exponents only. $$\left(w^{-7}\right)^{2}\left(w^{3}\right)^{4}$$

Simplify the expression. $$\left(2 p^{2}\right)^{3}(p q)^{2}$$
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