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Chapter 10: Problem 74

Simplify. Write the answers with positive exponents only. $$\frac{2^{3}}{2^{-2}}$$

Short Answer

Expert verified
32

Step by step solution

01

Apply the Law of Exponents

Use the law of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \) to combine the terms. In this case, \( \frac{2^3}{2^{-2}} \) can be written as \( 2^{3-(-2)} \).
02

Simplify the Exponent

Next, simplify the exponent. \( 3-(-2) \) becomes \( 3 + 2 = 5 \).
03

Write the Simplified Expression

The expression is now simplified to \( 2^5 \).
04

Calculate the Power

Finally, calculate \( 2^5 \). \( 2^5 = 32 \).

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Key Concepts

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

Law of Exponents
The law of exponents is a fundamental rule in algebra that helps in simplifying expressions involving powers. This law states that when two exponents with the same base are divided, you subtract the exponent of the denominator from the exponent of the numerator. For example, in the expression \( \frac{2^3}{2^{-2}} \), we use the rule: \( \frac{a^m}{a^n} = a^{m-n} \).
Here, both the numerator and the denominator have the same base (2), so we subtract the exponent in the denominator (which is -2) from the exponent in the numerator (which is 3). This results in the new exponent: \( 3 - (-2) \).
Positive Exponents
Positive exponents indicate how many times a number is multiplied by itself. In simplifying expressions, it is important to convert negative exponents into positive ones because it makes the calculations easier to interpret.
For instance, in the fraction \( \frac{2^3}{2^{-2}} \), we convert the negative exponent in the denominator by following the law of exponents. When we simplify \( 2^{3-(-2)} \), we get \( 2^5 \), which now has a positive exponent.
This makes it easier to further simplify or calculate the value.
Simplifying Expressions
Simplifying expressions involves rewriting them in a more compact, understandable form without changing their value. This often involves using algebraic rules such as the law of exponents. In our example, we start with \( \frac{2^3}{2^{-2}} \).
By applying \( 2^{3-(-2)} \), we simplify it to \( 2^5 \). This eliminates the fraction and the negative exponent, providing a clearer and simpler form.
Efficient simplification makes it easier to perform further calculations or evaluations.
Power Calculation
Power calculation involves raising a base number to an exponent. It is the result of multiplying the base by itself as many times as indicated by the exponent. In our example, we need to calculate \( 2^5 \).
This means multiplying 2 by itself 5 times: \( 2 \times 2 \times 2 \times 2 \times 2 \).
Performing the multiplication, we get: \( 2 \times 2 = 4 \), \( 4 \times 2 = 8 \), \( 8 \times 2 = 16 \), and finally, \( 16 \times 2 = 32 \).
Thus, \( 2^5 = 32 \). This final step provides the numerical value of the simplified expression.

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