Question

use the properties of exponents to determine the value of a for the equation below given x > 0
\\(\frac{x^{\frac{1}{5}}}{\sqrt5{x^3}} = x^a\\)

Explanation:

Step1: Rewrite the root as an exponent

The fifth root of \( x^3 \) can be written as \( x^{\frac{3}{5}} \) using the property \( \sqrt[n]{x^m}=x^{\frac{m}{n}} \). So the denominator \( \sqrt[5]{x^3} \) becomes \( x^{\frac{3}{5}} \).
The equation is now \( \frac{x^{\frac{1}{5}}}{x^{\frac{3}{5}}}=x^a \).

Step2: Use the quotient rule for exponents

The quotient rule for exponents states that \( \frac{x^m}{x^n}=x^{m - n} \) when \( x
eq0 \). Here, \( m = \frac{1}{5} \) and \( n=\frac{3}{5} \).
So, \( x^{\frac{1}{5}-\frac{3}{5}}=x^a \).

Step3: Simplify the exponent

Calculate \( \frac{1}{5}-\frac{3}{5} \). Since the denominators are the same, we subtract the numerators: \( \frac{1 - 3}{5}=\frac{-2}{5} \).
So we have \( x^{-\frac{2}{5}}=x^a \).

Step4: Equate the exponents

Since the bases are the same (\( x>0 \)) and the equation holds, the exponents must be equal. Therefore, \( a = -\frac{2}{5} \).

Answer:

\( -\frac{2}{5} \)