Simplify each exponential expression in Exercises 23–64. $(-4/x{)}^3$

Verified step by step guidance

1. Identify the expression to simplify: \(\left(\frac{-4}{x}\right)^3\).
2. Recall the exponent rule: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), which means you raise both numerator and denominator to the power of 3.
3. Apply the exponent to the numerator and denominator separately: \(\frac{(-4)^3}{x^3}\).
4. Calculate the power of the numerator: \((-4)^3 = (-4) \times (-4) \times (-4)\), which involves multiplying -4 three times.
5. Write the simplified expression as \(\frac{(-4)^3}{x^3}\), showing the numerator and denominator raised to the third power.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential for simplifying expressions like (−4/x)^3. This includes knowing how to apply the power of a product rule, which states that (ab)^n = a^n * b^n, and the power of a quotient rule, (a/b)^n = a^n / b^n.

Negative and Fractional Bases

When dealing with negative bases raised to powers, it is important to recognize how the exponent affects the sign. For odd exponents, the result remains negative, while for even exponents, it becomes positive. Also, understanding how to handle variables in the denominator is crucial.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions and applying exponent rules to both numerator and denominator. This includes raising both parts to the given power and simplifying the resulting expression to its simplest form.