Question

8.3 simplify rational exponents (homework) score: 19.05/25 answered: 20/25 question 21 rewrite using a rational exponent. assume all variables are positive. $sqrt5{3b^{6}y^{3}}=$ question help: e video d written example submit question jump to answer

Explanation:

Step1: Recall radical - exponent rule

The rule for converting a radical $\sqrt[n]{a}$ to a rational - exponent is $\sqrt[n]{a}=a^{\frac{1}{n}}$. For the expression $\sqrt[5]{3b^{6}y^{3}}$, we can rewrite it using this rule.
Here, $a = 3b^{6}y^{3}$ and $n = 5$. So, $\sqrt[5]{3b^{6}y^{3}}=(3b^{6}y^{3})^{\frac{1}{5}}$.

Step2: Apply the power - of - a - product rule

The power - of - a - product rule $(ab)^m=a^{m}b^{m}$. Applying this rule to $(3b^{6}y^{3})^{\frac{1}{5}}$, we get $3^{\frac{1}{5}}b^{6\times\frac{1}{5}}y^{3\times\frac{1}{5}}$.
Since $6\times\frac{1}{5}=\frac{6}{5}$ and $3\times\frac{1}{5}=\frac{3}{5}$, the expression becomes $3^{\frac{1}{5}}b^{\frac{6}{5}}y^{\frac{3}{5}}$.

Answer:

$3^{\frac{1}{5}}b^{\frac{6}{5}}y^{\frac{3}{5}}$