Problem 12 - College Algebra Essentials

Question

Evaluate each exponential expression. $$2^{-6}$$

Step-by-Step Solution

Answer

The evaluation of the expression \(2^{-6}\) yields \(\frac{1}{64}\).

Step 1: Understanding Negative Exponents

Any number to the power of a negative exponent can be expressed as one divided by that number to the power of the same positive exponent. This is mathematically expressed as \(a^{-n} = \frac{1}{a^n}\), where a is any real number and n is a positive integer. In this case, \(a = 2\) and \(n = 6\).

Step 2: Evaluating the Exponential Expression

Using the rule for negative exponents, \(2^{-6}\) can be rewritten as \(\frac{1}{2^6}\).

Step 3: Performing the Division

Now, calculate \(2^6\), which is \(64\). Substitute this into the denominator of \(\frac{1}{2^6}\), yielding \(\frac{1}{64}\).