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)/(p³)^-1/5

Verified step by step guidance

1. Rewrite the expression clearly: \(\frac{p^{\frac{1}{5}} \cdot p^{\frac{7}{10}} \cdot p^{\frac{1}{2}}}{(p^3)^{-\frac{1}{5}}}\).
2. Use the property of exponents for multiplication in the numerator: add the exponents of \(p\) since the bases are the same. So, calculate \(\frac{1}{5} + \frac{7}{10} + \frac{1}{2}\).
3. Simplify the denominator by applying the power of a power rule: \((p^3)^{-\frac{1}{5}} = p^{3 \times (-\frac{1}{5})} = p^{-\frac{3}{5}}\).
4. Rewrite the entire expression as \(p^{\text{(sum of numerator exponents)}} \div p^{-\frac{3}{5}}\), which is equivalent to \(p^{\text{(sum of numerator exponents)} - (-\frac{3}{5})}\).
5. Simplify the exponent by subtracting the negative exponent in the denominator, which is the same as adding the positive exponent, and write the final expression with a positive exponent.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include multiplying exponents when bases are the same and dividing exponents when dividing like bases. For example, a^m * a^n = a^(m+n) and (a^m)^n = a^(m*n). These rules help combine and simplify terms efficiently.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For instance, a^(-n) = 1/a^n. When simplifying expressions, negative exponents should be rewritten as positive exponents by moving the base between numerator and denominator, especially when the problem specifies no negative exponents.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, a^(m/n) means the n-th root of a raised to the m-th power, or (√[n]{a})^m. Understanding fractional exponents allows simplification of expressions involving roots and powers in a unified way.