Problem 598 Simplify. $$\left(k^{-2}\right... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 598

Simplify. $$\left(k^{-2}\right)^{-3}$$

Short Answer

Expert verified

The simplified form is $k^{6}$.

Step by step solution

Apply the Power Rule

The power rule for exponents states that $(a^m)^n = a^{m \times n}$. Apply this rule to the given expression $(k^{-2})^{-3}$.

Multiply the Exponents

Multiply the exponents: $-2 \times -3$. This yields $k^{6}$.

Rewrite the Simplified Form

After multiplying the exponents, the expression simplifies to $k^{6}$.

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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 simplifying expressions with exponents, one useful tool is the power rule for exponents. The power rule states that \[\begin{equation} (a^m)^n = a^{m \times n} \end{equation}\].
This means that when you raise a power to another power, you multiply the exponents together.

For example, in the expression \[\begin{equation} (k^{-2})^{-3} \end{equation}\], we apply the power rule. Here, we need to multiply the exponents \[\begin{equation} -2 \end{equation}\] and \[\begin{equation} -3 \end{equation}\] together.

Understanding this rule helps us to simplify complex expressions quickly and is an essential tool in algebra.

Negative Exponents

Negative exponents can be tricky, but they follow a straightforward rule: \[\begin{equation} a^{-m} = \frac{1}{a^m} \end{equation}\].
It means that a negative exponent indicates the reciprocal of the base raised to the positive exponent.

In our given expression \[\begin{equation} (k^{-2})^{-3} \end{equation}\], the exponent is negative.
However, applying the power rule first makes things simpler. Multiply \[\begin{equation} -2 \end{equation}\] by \[\begin{equation} -3 \end{equation}\], resulting in \[\begin{equation} k^{6} \end{equation}\] 鈥� a positive exponent.

Multiplying Exponents

In the realm of exponents, multiplying exponents is simple when you know the rules.

According to the power rule, we multiply exponents when raising one power to another. For the expression \[\begin{equation} (k^{-2})^{-3} \end{equation}\], this means we compute \[\begin{equation} -2 \times -3 \end{equation}\], which equals \[\begin{equation} 6 \end{equation}\].
As a result, our expression simplifies to \[\begin{equation} k^6 \end{equation}\].

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91影视

Simplify. $$\left(k^{-2}\right)^{-3}$$

Short Answer

Step by step solution

Apply the Power Rule

Multiply the Exponents

Rewrite the Simplified Form

Key Concepts

Power Rule for Exponents

Negative Exponents

Multiplying Exponents

One App. One Place for Learning.

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