Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. r⁸( —— )³ s²

Verified step by step guidance

1. Start with the given expression: \(\left( \frac{r^{8}}{s^{2}} \right)^{3}\).
2. Apply the power of a quotient rule, which states that \(\left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}}\). So rewrite the expression as \(\frac{\left(r^{8}\right)^{3}}{\left(s^{2}\right)^{3}}\).
3. Next, apply the power of a power rule, which states that \(\left(a^{m}\right)^{n} = a^{m \times n}\). Simplify the numerator to \(r^{8 \times 3} = r^{24}\) and the denominator to \(s^{2 \times 3} = s^{6}\).
4. Rewrite the expression as \(\frac{r^{24}}{s^{6}}\).
5. Since all variables represent nonzero real numbers, this is the simplified form of the expression.

Key Concepts

Exponentiation of a Quotient

When raising a quotient to a power, apply the exponent to both the numerator and the denominator separately. For example, (a/b)^n = a^n / b^n. This rule helps simplify expressions involving powers of fractions.

Power of a Power Rule

The power of a power rule states that (a^m)^n = a^(m*n). This means when an exponent is raised to another exponent, multiply the exponents. This rule is essential for simplifying expressions with nested exponents.

Properties of Exponents with Variables

Variables raised to powers follow the same exponent rules as numbers. When simplifying, treat variables as bases and apply exponent rules consistently, ensuring to keep track of positive and negative exponents, especially when variables are in denominators.