Simplify each exponential expression in Exercises 23–64. (−4/x)^3

Verified step by step guidance

1. Identify the base and the exponent in the expression \((-\frac{4}{x})^3\). The base is \(-\frac{4}{x}\) and the exponent is 3.
2. Apply the power of a quotient rule, which states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). This means \((-\frac{4}{x})^3 = \frac{(-4)^3}{x^3}\).
3. Calculate \((-4)^3\) by multiplying \(-4\) by itself three times: \(-4 \times -4 \times -4\).
4. Simplify the numerator \((-4)^3\) to get the result.
5. The expression is now simplified to \(\frac{(-4)^3}{x^3}\).

Key Concepts

Exponential Expressions

Exponential expressions involve a base raised to a power, indicating how many times the base is multiplied by itself. In the expression (−4/x)^3, the base is (−4/x) and the exponent is 3, meaning the base will be multiplied by itself three times. Understanding how to manipulate these expressions is crucial for simplification.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, x^(-n) is equivalent to 1/(x^n). In the expression (−4/x)^3, recognizing that x in the denominator can be treated with a negative exponent will aid in simplifying the expression correctly.

Distributive Property

The distributive property allows us to multiply a single term by each term within a parenthesis. When simplifying (−4/x)^3, applying the distributive property helps in expanding the expression to separate the base and the exponent, leading to clearer simplification steps. This property is essential for handling expressions with multiple components.