Question

8.3 simplify rational exponents (homework)
score: 23.05/25 answered: 24/25
question 25
simplify by writing as a single exponential expression. assume all variables are positive.
\\(\frac{y^{\frac{1}{3}}y^{-\frac{2}{3}}}{y^{\frac{5}{2}}}=\\)

Explanation:

Step1: Use exponent - product rule

When multiplying exponential expressions with the same base $a^m\cdot a^n=a^{m + n}$. So, $y^{\frac{1}{3}}y^{-\frac{2}{3}}=y^{\frac{1}{3}+(-\frac{2}{3})}=y^{\frac{1 - 2}{3}}=y^{-\frac{1}{3}}$.

Step2: Use exponent - quotient rule

When dividing exponential expressions with the same base $\frac{a^m}{a^n}=a^{m - n}$. So, $\frac{y^{-\frac{1}{3}}}{y^{\frac{5}{2}}}=y^{-\frac{1}{3}-\frac{5}{2}}$.

Step3: Find a common denominator

The common denominator of 3 and 2 is 6. Then $-\frac{1}{3}-\frac{5}{2}=-\frac{1\times2}{3\times2}-\frac{5\times3}{2\times3}=-\frac{2}{6}-\frac{15}{6}=-\frac{2 + 15}{6}=-\frac{17}{6}$.

Answer:

$y^{-\frac{17}{6}}$