Problem 20 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$2^{-3} \cdot 2$$

Step-by-Step Solution

Answer

The value of the expression \(2^{-3} \cdot 2\) is 0.25.

Step 1: Identify the Expression Base and Power

From the expression \(2^{-3} \cdot 2\), '2' is identified as the base, and '-3' as the power. Here, -3 is a negative exponent.

Step 2: Apply the Negative Exponentiation Rule

According to the negative exponentiation rule \(a^{-n} = 1/a^n\), the sub-expression \(2^{-3}\) can be rewritten as \(1/2^3 = 1/8\).

Step 3: Perform the Multiplication

Now, multiply the result from Step 2 by 2: \(2 \cdot \frac{1}{8} = 0.25\)

Key Concepts

Understanding Negative Exponents

Negative exponents might seem intimidating at first, but they follow a simple rule. Whenever you have a negative exponent, like in the expression \(2^{-3}\), the rule is to invert the base and change the negative exponent to a positive. So, \(a^{-n}\) becomes \(1/a^n\). This means for \(2^{-3}\), you rewrite it as \(1/2^3\), which equals \(1/8\). Understanding this rule helps in simplifying expressions and making calculations easier. By converting a negative exponent to a fraction, you are effectively changing the operation from repeated multiplication to division by the same factor.
Furthermore, this concept of negative exponents is crucial when solving problems because it indicates how many times the base is divided, rather than multiplied, making computations involving negative exponents simpler to handle.

Comprehending Base and Power

In any exponential expression like \(2^{-3}\), identifying the base and power is essential. The **base** is the number that is being multiplied, and the **power** indicates how many times the base is used as a factor. In our situation, '2' is the base, and '-3' is the power. This means that without considering the negative sign, '2' would be multiplied by itself three times.
However, because the power is negative, instead of multiplying, you divide when simplifying \(2^{-3}\).

• The **base** provides the value that gets repeatedly multiplied or divided.
• The **power** dictates the number of times the operation occurs and determines whether it's multiplication or division based on the sign.

Understanding the base and power in exponential expressions is critical. It forms the basis for applying rules and executing calculations correctly. Always remember, the base is the constant number and the power tells you the number of times to multiply or divide the base.

Multiplication in Exponents

Multiplying exponential expressions involves using known rules to simplify and solve the problem. When dealing with issues such as \(2^{-3} \cdot 2\), you must apply the rewritten result of any negative exponent before proceeding with additional operations. This ensures each part of the expression is correctly simplified before final multiplying.
Initially, convert \(2^{-3}\) to \(1/8\) based on the negative exponent rule. Subsequently, apply multiplication, as follows:

• Multiply \(2 \cdot \frac{1}{8}\).
• Calculate the result \((2 \cdot \frac{1}{8} = 0.25)\).

Multiplication in exponents often requires stringing together multiple rules, such as those for negative exponents, to reach a complete solution. Steps should be clear and orderly to maintain accuracy throughout the calculations. This ensures that students appreciate how to manage exponents systematically, leading to better problem-solving skills when tackling more complex expressions.