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Chapter 2: Problem 20

Rewrite \(\left(5^{-2}\right)^{-4}\). A. \(\frac{1}{5^8}\) B. \(5^8\) C. 400 D. 10,000

Short Answer

Expert verified
B. \( 5^8 \)

Step by step solution

01

Apply the Power of a Power Rule

Use the power of a power rule, which states that \((a^m)^n = a^{m \times n}\), to rewrite \( (5^{-2})^{-4} \). This yields \( 5^{(-2) \times (-4)} \).
02

Simplify the Exponent

Calculate the exponent by multiplying -2 and -4: \(-2 \times -4 = 8\). Thus, \( 5^{(-2) \times (-4)} = 5^8 \).
03

Compare with the Provided Options

The expression \( 5^8 \) matches with option B.

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Key Concepts

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

Power of a Power Rule
The Power of a Power Rule helps simplify expressions where an exponent is raised to another exponent. This rule states that \( (a^m)^n = a^{m \times n} \). This means you multiply the exponents together.
For example, let's apply this to our exercise: \( (5^{-2})^{-4} \). By the Power of a Power Rule, we multiply the exponents: \( -2 \times -4 \). This results in \( 5^{8} \).
Key Points:
  • Only multiply the exponents.
  • Do not alter the base.
  • Remember to handle the negative signs correctly.

This is a very effective rule when simplifying nested exponents.
Simplifying Exponents
Now that we've applied the Power of a Power Rule, we need to simplify our expression.
Simplifying exponents involves performing the mathematical operations on the exponents themselves. In our exercise, we had \( 5^{-2} \) raised to the power of \( -4 \. \) So: \( -2 \times -4 = 8 \).
This leaves us with \( 5^{8} \. \) Key Points:
  • Follow the rules of multiplication or division as required.
  • Treat negative exponents just like positive ones when multiplying or dividing.
  • When you multiply two negative numbers, the result is always positive.

With these steps, we can confirm that \( 5^{-2} \) raised to the power of \(-4 \) simplifies to \( 5^{8} \).
Mathematical Reasoning
Mathematical reasoning involves logical steps to solve a problem. You need to understand the rules and how to apply them.
For this exercise, the reasoning goes as follows:
  • Identify which exponential rule applies (Power of a Power Rule).
  • Apply the rule to rewrite the expression.
  • Simplify the exponents following standard multiplication rules.
  • Cross-check the simplified expression with the given options.

Proper mathematical reasoning helps in understanding why we are doing each step. In this case, understanding why \( (5^{-2})^{-4} \) becomes \( 5^{8} \) clarifies how the rules come together. As a result, you can compare \( 5^8 \) to the provided choices and conclude that the correct answer is option B.

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Most popular questions from this chapter

Compute \(4^3\). A. 12 B. 81 C. 64 D. 256

Compute \(2^6 \cdot 5^6\) A. 360 B. 60 C. \(1,000,000\) D. 420

Select the expression equivalent to \(2(3+5-4)\). A. \(5+1\) B. \(6+1\) C. \(5+7-6\) D. \(6+10-8\)

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Evaluate \(25^{-\frac{1}{2}}\) A. \(-\frac{25}{2}\) B. \(\frac{1}{625}\) C. \(\frac{1}{5}\) D. -625
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