Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (p^1/5p^7/10p^1/2)/(p³)^-1/5

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{p^{\frac{1}{5}} p^{\frac{7}{10}} p^{\frac{1}{2}}}{\left(p^{3}\right)^{-\frac{1}{5}}}\).
2. Apply the product rule of exponents to the numerator by adding the exponents: \(p^{\frac{1}{5} + \frac{7}{10} + \frac{1}{2}}\).
3. Simplify the exponent sum in the numerator by finding a common denominator and adding: \(\frac{1}{5} + \frac{7}{10} + \frac{1}{2}\).
4. Simplify the denominator by applying the power of a power rule: \(\left(p^{3}\right)^{-\frac{1}{5}} = p^{3 \times -\frac{1}{5}}\).
5. Rewrite the entire expression as a single power of \(p\) by subtracting the exponent in the denominator from the exponent in the numerator, and then express the answer without negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include adding exponents when multiplying like bases, subtracting exponents when dividing, and multiplying exponents when raising a power to another power. These rules allow combining and simplifying expressions with variables raised to fractional or negative powers.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-a) = 1/x^a. When simplifying, expressions with negative exponents should be rewritten without negatives by moving factors between numerator and denominator, especially important when the problem specifies no negative exponents.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For instance, x^(m/n) means the n-th root of x raised to the m-th power. Understanding how to add, subtract, and multiply fractional exponents is essential for simplifying expressions involving variables raised to fractional powers.