Question

8.3 simplify rational exponents (homework)
score: 18.05/25 answered: 19/25
question 20
rewrite using a rational exponent. assume all variables are positive.
$sqrt3{\frac{2b^{5}}{y^{7}}}=$

Explanation:

Step1: Recall radical - exponent rule

The rule for converting a radical $\sqrt[n]{a}$ to a rational - exponent is $a^{\frac{1}{n}}$. For the given expression $\sqrt[3]{\frac{2b^{5}}{y^{7}}}$, we can rewrite it as $(\frac{2b^{5}}{y^{7}})^{\frac{1}{3}}$.

Step2: Apply power - of - a - quotient rule

The power - of - a - quotient rule $( \frac{a}{b})^n=\frac{a^n}{b^n}$. So, $(\frac{2b^{5}}{y^{7}})^{\frac{1}{3}}=\frac{2^{\frac{1}{3}}(b^{5})^{\frac{1}{3}}}{(y^{7})^{\frac{1}{3}}}$.

Step3: Apply power - of - a - power rule

The power - of - a - power rule $(a^{m})^{n}=a^{mn}$. So, $(b^{5})^{\frac{1}{3}} = b^{\frac{5}{3}}$ and $(y^{7})^{\frac{1}{3}}=y^{\frac{7}{3}}$. Then the expression becomes $\frac{2^{\frac{1}{3}}b^{\frac{5}{3}}}{y^{\frac{7}{3}}}$.

Answer:

$\frac{2^{\frac{1}{3}}b^{\frac{5}{3}}}{y^{\frac{7}{3}}}$