Simplify each expression. Assume all variables represent nonzero real numbers. (-5n⁴/r²)³

Verified step by step guidance

1. Identify the expression to simplify: \(\left( \frac{-5n^{4}}{r^{2}} \right)^{3}\).
2. Apply the power of a quotient rule: \(\left( \frac{a}{b} \right)^{m} = \frac{a^{m}}{b^{m}}\). So rewrite the expression as \(\frac{(-5n^{4})^{3}}{(r^{2})^{3}}\).
3. Apply the power of a product rule to the numerator: \((-5n^{4})^{3} = (-5)^{3} \cdot (n^{4})^{3}\).
4. Simplify the powers inside the numerator: \((-5)^{3}\) remains as is, and use the power of a power rule \( (n^{4})^{3} = n^{4 \times 3} = n^{12}\).
5. Simplify the denominator using the power of a power rule: \((r^{2})^{3} = r^{2 \times 3} = r^{6}\). Now the expression is \(\frac{(-5)^{3} n^{12}}{r^{6}}\).

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 Product Rule

Raising a product to a power means raising each factor to that power individually: (ab)^n = a^n * b^n. This allows you to distribute the exponent across all factors inside parentheses.

Exponent Rules for Variables

When raising a variable with an exponent to another power, multiply the exponents: (x^m)^n = x^(m*n). This rule simplifies expressions with variables raised to powers raised to powers.