Problem 12 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$2^{-6}$$

Step-by-Step Solution

Answer

The value of \(2^{-6}\) is \(\frac{1}{64}\).

Step 1: Understanding Negative Exponent Rule

The rule states that any non zero number \(a\) raised to a negative exponent \(b\) is equivalent to 1 divided by the number \(a\) raised to the positive exponent \(b\). Symbolically, it can be represented as: \(a^{-b} = \frac{1}{a^b}\).

Step 2: Apply Negative Exponent Rule to the Given Expression

Substitute \(a\) with 2 and \(b\) with 6 in the rule to get \(2^{-6} = \frac{1}{2^6}\).

Step 3: Evaluating the Expression

Now, calculate \(2^6\), which is equal to 64 and substitute in the denominator, thus we get \(2^{-6} = \frac{1}{64}\).

Key Concepts

Exponential Expressions

Exponential expressions, such as \(2^{-6}\), are a way to represent repeated multiplication of a number by itself. In other words, an expression like \(a^n\) indicates that the base \(a\) should be multiplied by itself \(n\) times.

For example, \(2^3\) would be calculated as \(2 \times 2 \times 2\), which equals 8. When dealing with negative exponents, it's essential to understand that they don't imply negative numbers but are a notation for expressing reciprocals. Specifically, a negative exponent indicates that the base should be taken to the corresponding positive exponent, and then we take the reciprocal of that number. This concept is critical for simplifying and evaluating expressions where variables or constants are raised to negative exponents.

Evaluating Exponents

Evaluating exponents refers to finding the value of expressions involving exponents. For positive exponents, this is straightforward as it involves multiplying the base by itself the specified number of times. In contrast, evaluating negative exponents requires us to use the negative exponent rule. According to this rule, any base \(a\) other than zero raised to a negative exponent \(-b\) is equal to \(1\) divided by \(a\) raised to the positive exponent \(b\), or written algebraically as \(a^{-b} = \frac{1}{a^b}\).

In practice, to evaluate \(2^{-6}\), you first calculate \(2^6\), which is the base multiplied by itself six times. After finding the value of \(64\), the reciprocal is taken to get the final answer of \(\frac{1}{64}\). This method transforms the process of evaluation into a simple, stepwise calculation.

Algebraic Rules

Algebraic rules are the established mathematical laws that govern how to manipulate and simplify algebraic expressions. One such rule is the negative exponent rule. This rule assists in simplifying expressions and solving equations that involve exponents and can be applied to both numerical bases as well as algebraic variables.

The application of the negative exponent rule is not limited to simple constants; it's equally important when expressions become more complex with variables. For instance, if you have \(x^{-n}\), you would rewrite this as \(\frac{1}{x^n}\) according to the rule. By mastering this and other algebraic rules, students can approach a broad range of problems with confidence and versatility, enhancing their problem-solving skills in algebra and beyond.