Question

8.3 simplify rational exponents (homework) score: 13.05/25 answered: 14/25 question 15 rewrite using a rational exponent. assume all variables are positive. $sqrt7{\frac{y^{5}}{x^{8}}}=$ question help: video written example

Explanation:

Step1: Recall radical - exponent rule

The rule for converting a radical $\sqrt[n]{a}$ to an exponent is $a^{\frac{1}{n}}$. For the given expression $\sqrt[7]{\frac{y^{5}}{x^{8}}}$, we can rewrite it as $(\frac{y^{5}}{x^{8}})^{\frac{1}{7}}$.

Step2: Apply power - of - a - quotient rule

The power - of - a - quotient rule $( \frac{a}{b})^m=\frac{a^{m}}{b^{m}}$. So, $(\frac{y^{5}}{x^{8}})^{\frac{1}{7}}=\frac{y^{5\times\frac{1}{7}}}{x^{8\times\frac{1}{7}}}$.

Step3: Simplify exponents

$5\times\frac{1}{7}=\frac{5}{7}$ and $8\times\frac{1}{7}=\frac{8}{7}$. So the expression becomes $\frac{y^{\frac{5}{7}}}{x^{\frac{8}{7}}}$.

Answer:

$\frac{y^{\frac{5}{7}}}{x^{\frac{8}{7}}}$