Evaluate each exponential expression in Exercises 1–22. 4^−3

Verified step by step guidance

1. Identify the base and the exponent in the expression. Here, the base is 4 and the exponent is -3.
2. Recall the property of exponents that states: $a^{-n}=\frac{1}{a^n}$. This means that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
3. Apply this property to the expression $4^{-3}$, which becomes $\frac{1}{4^3}$.
4. Calculate $4^3$ by multiplying 4 by itself three times: $4\times 4\times 4$.
5. Express the final result as $\frac{1}{\text{the value of }{4}^3}$.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical expressions that involve a base raised to a power, represented as a^b, where 'a' is the base and 'b' is the exponent. The exponent indicates how many times the base is multiplied by itself. Understanding how to evaluate these expressions is crucial for solving problems involving growth, decay, and other phenomena modeled by exponential functions.

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. For example, a^−b is equivalent to 1/(a^b). This concept is essential for simplifying expressions and understanding how to manipulate exponential terms, especially when dealing with fractions or inverse relationships.

Evaluating Exponents

Evaluating exponents involves calculating the value of an exponential expression by applying the rules of exponents. This includes multiplying the base by itself as many times as indicated by the exponent, and applying the rules for negative and zero exponents. Mastery of this process is necessary for accurately solving problems that involve exponential growth or decay.