Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (p^3)^1/4/(p^5/4)^2

Verified step by step guidance

1. Start by simplifying the expression \((p^3)^{1/4}\). Use the power of a power property \((a^m)^n = a^{m \cdot n}\) to rewrite it as \(p^{3 \cdot \frac{1}{4}} = p^{3/4}\).
2. Next, simplify the expression \((p^{5/4})^2\). Again, use the power of a power property to rewrite it as \(p^{5/4 \cdot 2} = p^{5/2}\).
3. Now, you have the expression \(\frac{p^{3/4}}{p^{5/2}}\). Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to combine the exponents: \(p^{3/4 - 5/2}\).
4. Convert the exponents to have a common denominator. Rewrite \(3/4\) as \(6/8\) and \(5/2\) as \(20/8\).
5. Subtract the exponents: \(p^{6/8 - 20/8} = p^{-14/8}\). Simplify the exponent to \(p^{-7/4}\) and rewrite it without negative exponents as \(\frac{1}{p^{7/4}}\).

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that govern how to manipulate expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the quotient of powers (a^m / a^n = a^(m-n)). Understanding these rules is essential for simplifying expressions with exponents.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). In the context of simplification, it is important to rewrite expressions without negative exponents, which often involves moving terms from the numerator to the denominator or vice versa.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and can be expressed in terms of exponents. For instance, the nth root of a number can be represented as a^(1/n). When simplifying expressions with fractional exponents, it is crucial to recognize how to convert between radical and exponential forms to achieve a simplified result.