Question

simplify.
9^{\frac{1}{3}}cdot3^{\frac{1}{3}}

Explanation:

Step1: Rewrite 9 as 3^2

$9^{\frac{1}{3}}\cdot3^{\frac{1}{3}}=(3^2)^{\frac{1}{3}}\cdot3^{\frac{1}{3}}$

Step2: Apply power - of - a - power rule

According to the power - of - a - power rule $(a^m)^n=a^{mn}$, so $(3^2)^{\frac{1}{3}} = 3^{2\times\frac{1}{3}}=3^{\frac{2}{3}}$. Then the expression becomes $3^{\frac{2}{3}}\cdot3^{\frac{1}{3}}$.

Step3: Apply product rule of exponents

According to the product rule of exponents $a^m\cdot a^n=a^{m + n}$, so $3^{\frac{2}{3}}\cdot3^{\frac{1}{3}}=3^{\frac{2}{3}+\frac{1}{3}}$.

Step4: Calculate the exponent sum

$\frac{2}{3}+\frac{1}{3}=\frac{2 + 1}{3}=1$. So $3^{\frac{2}{3}+\frac{1}{3}}=3^1 = 3$.

Answer:

3