Evaluate each expression without using a calculator. 16^-6/2

Verified step by step guidance

1. Recognize that the expression is an exponentiation with a negative fractional exponent: \(16^{\left(-\frac{6}{2}\right)}\).
2. Simplify the exponent by dividing the numerator by the denominator: \(-\frac{6}{2} = -3\), so the expression becomes \$16^{-3}$.
3. Recall that a negative exponent means taking the reciprocal: \(a^{-n} = \frac{1}{a^n}\). So, rewrite \$16^{-3}$ as \(\frac{1}{16^3}\).
4. Express 16 as a power of 2, since 16 is \$2^4\(, so \)16^3\( becomes \)(2^4)^3$.
5. Use the power of a power rule: \((a^m)^n = a^{m \times n}\), so \((2^4)^3 = 2^{4 \times 3} = 2^{12}\). Therefore, the expression is \(\frac{1}{2^{12}}\).

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^(-n) = 1 / a^n. This concept allows rewriting expressions with negative powers into fractions for easier evaluation.

Exponentiation of Powers

When raising a power to another power, multiply the exponents: (a^m)^n = a^(m*n). This rule simplifies expressions like 16^(-6/2) by combining the exponent terms before evaluating the base.

Simplifying Base Numbers

Expressing the base as a power of a smaller number can simplify calculations. For example, 16 can be written as 2^4, which helps in applying exponent rules more easily and evaluating the expression without a calculator.