Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (p^1/5p^7/10p^1/2)/(p^3)^-1/5

Verified step by step guidance

1. First, simplify the expression in the numerator: \( p^{1/5} \cdot p^{7/10} \cdot p^{1/2} \).
2. Combine the exponents in the numerator by adding them: \( \frac{1}{5} + \frac{7}{10} + \frac{1}{2} \).
3. Convert all fractions to have a common denominator and add them together.
4. Simplify the expression in the denominator: \( (p^3)^{-1/5} \) which is equivalent to \( p^{-3/5} \).
5. Divide the simplified numerator by the simplified denominator by subtracting the exponents: \( p^{\text{numerator exponent} - (-3/5)} \).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power (a^(m*n) = a^mn). These rules help in combining and simplifying terms effectively.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent (a^-n = 1/a^n). In simplification, it's important to convert negative exponents to positive ones to adhere to the problem's requirement of expressing answers without negative exponents, which often involves rewriting the expression.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process often requires applying exponent rules and ensuring that all variables are treated as positive real numbers, which affects how we handle the exponents in the expression.