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Chapter 2: Problem 19

Simplify \(\left(2^{-3}\right)^2\). A. 36 B. \(\frac{1}{32}\) C. \(\frac{1}{64}\) D. -12

Short Answer

Expert verified
C. \( \frac{1}{64} \)

Step by step solution

01

Apply the power rule for exponents

The power rule for exponents states that \( (a^m)^n = a^{mn} \). Apply this rule to the given expression \( \big(2^{-3}\big)^2 \). So, \( (2^{-3})^2 = 2^{(-3) \times 2} \).
02

Multiply the exponents

Multiply the exponents -3 and 2: \( (-3) \times 2 = -6 \). Thus, the expression becomes \( 2^{-6} \).
03

Simplify the exponent

Recall that a negative exponent means the reciprocal of the base raised to the positive exponent. Therefore, \( 2^{-6} = \frac{1}{2^6} \).
04

Calculate the power of 2

Calculate \( 2^6 \), which is \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \). So, \( 2^{-6} = \frac{1}{64} \).

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Key Concepts

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

Power Rule for Exponents
The power rule for exponents is a fundamental concept in algebra. It is used to simplify expressions where a power is raised to another power. The rule states that \[ (a^m)^n = a^{mn} \].

This means that when you have an exponent raised to another exponent, you can multiply the two exponents together. For example, if you have \( (2^{-3})^2 \), you apply the rule as follows: \[ (2^{-3})^2 = 2^{(-3) \times 2} = 2^{-6} \].

Remember: The bases must be the same when applying this rule. This makes simplifying expressions much easier and helps you in solving more complex problems in algebra.
Negative Exponents
Negative exponents are another crucial concept to understand in algebra. A negative exponent indicates the reciprocal of the base raised to the positive exponent.

In mathematical terms, \[ a^{-n} = \frac{1}{a^n} \].

For instance, in our problem, we simplified \( 2^{-6} \) to its reciprocal form. This becomes \[ 2^{-6} = \frac{1}{2^6} \].

Understanding negative exponents helps you to transition smoothly between different forms of expressions and simplifies complex calculations. It's particularly useful when dealing with scientific notation, fractions, and solving equations.
Calculation of Powers
Calculating powers involves multiplying the base by itself as many times as the exponent indicates. For example, \( 2^6 \) means you multiply 2 six times. So the calculation would be: \[ 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \].

In the context of our problem, after applying the power rule for exponents and simplifying the negative exponent, we arrived at the expression \( \frac{1}{2^6} \).

We calculated \( 2^6 \) to find that it equals 64. Therefore, \( 2^{-6} = \frac{1}{64} \).

This process shows how understanding and correctly applying the rules for exponents can simplify even seemingly complex expressions.

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Most popular questions from this chapter

Rewrite \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) with an exponent. A. \(2^5\) B. \(5^2\) C. 32 D. 10

What is \(\sqrt[3]{64} ?\) A. 21 B. 8 C. 6 D. 4

Compute \(4^{2^{-1}}\). A. 2 B. 16 C. -16 D. -2

Simplify \(7^3 \cdot 7^2 \cdot 7^5\). A. \(7^{30}\) B. \(7^{10}\) C. \(343^{30}\) D. \(343^{10}\)

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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