In Exercises 83–90, evaluate each expression without using a calculator. 16^(−6/2)

Verified step by step guidance

1. Recognize that the expression is a power of a power: ${16}^{-\frac{6}{2}}$.
2. Simplify the exponent $-\frac{6}{2}$ to $-3$.
3. Rewrite the expression as ${16}^{-3}$.
4. Recall that a negative exponent indicates the reciprocal: ${16}^{-3}=\frac{1}{{16}^3}$.
5. Express ${16}^3$ as $16\times 16\times 16$ and simplify further if needed.

Key Concepts

Exponents and Powers

Exponents represent repeated multiplication of a base number. For example, in the expression a^n, 'a' is the base and 'n' is the exponent, indicating that 'a' is multiplied by itself 'n' times. Understanding how to manipulate exponents, including negative and fractional exponents, is crucial for evaluating expressions like 16^(-6/2).

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For instance, a^(-n) equals 1/(a^n). This concept is essential for simplifying expressions with negative exponents, such as converting 16^(-6/2) into a more manageable form.

Fractional Exponents

Fractional exponents represent roots in addition to powers. The expression a^(m/n) can be interpreted as the n-th root of a raised to the m-th power. In the case of 16^(-6/2), the exponent can be simplified to -3, which involves both the concept of negative exponents and the understanding of roots, as 16 can be expressed as 2^4.