Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (p³)^1/4/(p^5/4)²

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\frac{(p^{3})^{\frac{1}{4}}}{(p^{\frac{5}{4}})^{2}}\).
2. Apply the power of a power rule, which states that \((a^{m})^{n} = a^{m \times n}\), to both the numerator and the denominator: numerator becomes \(p^{3 \times \frac{1}{4}}\) and denominator becomes \(p^{\frac{5}{4} \times 2}\).
3. Simplify the exponents by multiplying: numerator exponent is \(\frac{3}{4}\) and denominator exponent is \(\frac{10}{4}\) (which can be simplified further if desired).
4. Rewrite the expression as a single power of \(p\) by subtracting the exponent in the denominator from the exponent in the numerator: \(p^{\frac{3}{4} - \frac{10}{4}}\).
5. Simplify the exponent subtraction and rewrite the expression without negative exponents by using the property \(a^{-m} = \frac{1}{a^{m}}\) if necessary.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers, such as multiplying powers with the same base by adding exponents, raising a power to another power by multiplying exponents, and dividing powers by subtracting exponents. These rules are essential for simplifying expressions like (p^3)^(1/4) and (p^(5/4))^2.

Negative Exponents and Their Conversion

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, p^(-n) = 1/p^n. Understanding how to rewrite expressions without negative exponents is crucial, as the problem requires answers without negative exponents, ensuring all terms have positive powers.

Simplifying Radical and Fractional Exponents

Fractional exponents represent roots, where the denominator is the root and the numerator is the power, such as p^(1/4) meaning the fourth root of p. Simplifying expressions with fractional exponents involves applying exponent rules carefully to combine and reduce terms.