Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. (p^-2)⁰/5p^-4

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{(p^{-2})^{0}}{5p^{-4}}\).
2. Apply the zero exponent rule to the numerator: any nonzero base raised to the zero power equals 1, so \((p^{-2})^{0} = 1\).
3. Rewrite the expression as \(\frac{1}{5p^{-4}}\).
4. Use the property of negative exponents to rewrite \(p^{-4}\) as \(\frac{1}{p^{4}}\), so the denominator becomes \(5 \times \frac{1}{p^{4}}\).
5. Simplify the denominator to \(\frac{5}{p^{4}}\), then rewrite the entire expression as \(1 \div \frac{5}{p^{4}}\), which equals \(\frac{p^{4}}{5}\). This expression has no negative exponents.

Key Concepts

Zero Exponent Rule

Any nonzero base raised to the zero power equals 1. For example, p^0 = 1 regardless of the value of p (as long as p ≠ 0). This rule simplifies expressions where exponents are zero.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For instance, p^(-n) = 1/p^n. Expressions should be rewritten without negative exponents by moving factors between numerator and denominator.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves applying exponent rules and combining like terms to write expressions in simplest form. This includes reducing powers, eliminating negative exponents, and ensuring the expression is clear and concise.