Question

In Exercises \(5-8,\) rewrite the exponential expression to have the indicated base. \((1 / 8)^{2 x}, \quad\) base 2

Short answer

The exponential expression \((1/8)^{2x}\) with base 2 is \(2^{-6x}\).

Step-by-step solution

Rewriting \(1/8\)

Recognize that \(1/8 = 2^{-3}\). This is a fundamental relationship that will be used to change the base of the exponential expression from \(1/8\) to \(2\).

Substitution

Substitute the equivalent of \(1/8\) that we derived in the previous step into the original expression. Therefore, \((1/8)^{2x} = (2^{-3})^{2x}\).

Simplification

Simplify the expression by multiplying the exponents. The law of exponents states that when an exponent is raised to another exponent, the exponents are multiplied. Therefore, the simplified expression is \(2^{-3*2x} = 2^{-6x}\).