Question

8.3 simplify rational exponents (homework) score: 21.05/25 answered: 22/25 question 23 simplify. assume all variables are positive. (8y^(9/2))^(2/3)= question help: video written example submit question jump to answer

Explanation:

Step1: Apply power - of - a - product rule

Use the rule $(ab)^n=a^n\times b^n$. So, $(8y^{\frac{9}{2}})^{\frac{2}{3}} = 8^{\frac{2}{3}}\times(y^{\frac{9}{2}})^{\frac{2}{3}}$.

Step2: Simplify $8^{\frac{2}{3}}$

Rewrite $8$ as $2^3$. Then $8^{\frac{2}{3}}=(2^3)^{\frac{2}{3}}$. By the power - of - a - power rule $(a^m)^n=a^{mn}$, we have $(2^3)^{\frac{2}{3}}=2^{3\times\frac{2}{3}} = 2^2=4$.

Step3: Simplify $(y^{\frac{9}{2}})^{\frac{2}{3}}$

Using the power - of - a - power rule $(a^m)^n=a^{mn}$, we get $(y^{\frac{9}{2}})^{\frac{2}{3}}=y^{\frac{9}{2}\times\frac{2}{3}}=y^3$.

Step4: Combine the results

$8^{\frac{2}{3}}\times(y^{\frac{9}{2}})^{\frac{2}{3}}=4y^3$.

Answer:

$4y^3$