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: \(\frac{(p^3)^{\frac{1}{4}}}{\left(p^{\frac{5}{4}}\right)^2}\).
2. Apply the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\), to both the numerator and the denominator: numerator becomes \(p^{3 \cdot \frac{1}{4}} = p^{\frac{3}{4}}\), denominator becomes \(p^{\frac{5}{4} \cdot 2} = p^{\frac{10}{4}}\).
3. Rewrite the expression now as \(\frac{p^{\frac{3}{4}}}{p^{\frac{10}{4}}}\).
4. Use the quotient rule for exponents, which says \(\frac{a^m}{a^n} = a^{m-n}\), to combine the powers of \(p\): \(p^{\frac{3}{4} - \frac{10}{4}}\).
5. Simplify the exponent by subtracting the fractions: \(\frac{3}{4} - \frac{10}{4} = -\frac{7}{4}\). Since the problem asks for no negative exponents, rewrite \(p^{-\frac{7}{4}}\) as \(\frac{1}{p^{\frac{7}{4}}}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include (a^m)^n = a^(m*n) and a^m / a^n = a^(m-n). Applying these rules allows combining and simplifying powers efficiently.

Negative Exponents

Negative exponents indicate reciprocals, where a^(-n) = 1/a^n. To write expressions without negative exponents, rewrite terms with negative powers as fractions with positive exponents in the denominator.

Fractional Exponents

Fractional exponents represent roots, such as a^(1/n) = n√a. Understanding how to manipulate fractional powers is essential for simplifying expressions like (p^3)^(1/4) and (p^(5/4))^2.