Question

simplify the expression.
\\(\frac{a^{\frac{1}{2}}a^{\frac{1}{3}}}{a^{\frac{1}{4}}}\\)
write your answer using only positive exponents.
assume that all variables are positive real numbers.

Explanation:

Step1: Add exponents in numerator

When multiplying terms with the same base, add exponents:
$a^{-\frac{1}{2}}a^{\frac{1}{3}} = a^{-\frac{1}{2}+\frac{1}{3}}$
Calculate the exponent:
$-\frac{1}{2}+\frac{1}{3} = -\frac{3}{6}+\frac{2}{6}=-\frac{1}{6}$
So the numerator simplifies to $a^{-\frac{1}{6}}$.

Step2: Subtract exponents for division

When dividing terms with the same base, subtract the denominator exponent from the numerator exponent:
$\frac{a^{-\frac{1}{6}}}{a^{\frac{1}{4}}} = a^{-\frac{1}{6}-\frac{1}{4}}$
Calculate the exponent:
$-\frac{1}{6}-\frac{1}{4} = -\frac{2}{12}-\frac{3}{12}=-\frac{5}{12}$

Step3: Convert to positive exponent

A term with a negative exponent is the reciprocal with a positive exponent:
$a^{-\frac{5}{12}} = \frac{1}{a^{\frac{5}{12}}}$

Answer:

$\frac{1}{a^{\frac{5}{12}}}$