Problem 20 Rewrite \(\left(5^{-2}\right)^{-... [FREE SOLUTION]

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Mathematical Reasoning Workbook for the GED Test

McGraw-Hill Education

$Math Studyset 91影视 Explanations$ Math

4 Edition

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

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)} $.

Simplify the Exponent

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

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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91影视

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

Short Answer

Step by step solution

Apply the Power of a Power Rule

Simplify the Exponent

Compare with the Provided Options

Key Concepts

Power of a Power Rule

Simplifying Exponents

Mathematical Reasoning

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