Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. (p⁴/q)²

Verified step by step guidance

1. Start with the given expression: \(\left( \frac{p^4}{q} \right)^2\).
2. Apply the exponent to both the numerator and the denominator inside the parentheses separately, using the rule \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\).
3. This gives \(\frac{(p^4)^2}{q^2}\).
4. Next, simplify the numerator by using the power of a power rule: \((p^4)^2 = p^{4 \times 2} = p^8\).
5. Write the simplified expression as \(\frac{p^8}{q^2}\).

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 like (p^4)^2.

Properties of Exponents with Variables

Variables with exponents follow the same rules as numbers. When simplifying expressions, treat variables as bases and apply exponent rules consistently. Assuming variables are nonzero avoids undefined expressions like division by zero.