Question

Find the simplified product ^3 sqrt 9x^4 * ^3 sqrt 3x^8

Answer 1

Answer:

$3x^4$

Step-by-step explanation:

$\sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8}$

To simplify it we multiply all the terms inside the cube root

$\sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8}$

$\sqrt[3]{9x^4 \cdot 3x^8}$

Now we apply exponential property

$a^m \cdot a^m = a^{mn}$

$x^4 \cdot x^8 = x^{12}$

$\sqrt[3]{9x^4 \cdot 3x^8}$

$\sqrt[3]{27x^{12}}$

Now we take cube root

$\sqrt[3]{27}=3$

$\sqrt[3]{x^{12}}=\sqrt[3]{x^3 \cdot x^3 \cdot x^3 \cdot x^3}=x^4$

$\sqrt[3]{27x^{12}}$

$3x^4$

Answer 2

$\bf ~\hspace{7em}\textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-\frac{ n}{ m}} \implies \cfrac{1}{a^{\frac{ n}{ m}}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \sqrt[3]{9x^4}\cdot \sqrt[3]{3x^8}\implies (9x^4)^{\frac{1}{3}}\cdot (3x^8)^{\frac{1}{3}}\implies 9^{\frac{1}{3}}\cdot x^{4\cdot \frac{1}{3}}\cdot 3^{\frac{1}{3}}\cdot x^{8\cdot \frac{1}{3}}$

$\bf 9^{\frac{1}{3}}\cdot 3^{\frac{1}{3}}\cdot x^{\frac{4}{3}}\cdot x^{\frac{8}{3}}\implies (3^2)^{\frac{1}{3}}\cdot 3^{\frac{1}{3}}\cdot x^{\frac{4}{3}+\frac{8}{3}}\implies 3^{\frac{2}{3}}\cdot 3^{\frac{1}{3}}\cdot x^{\frac{12}{3}} \\\\\\ 3^{\frac{2}{3}+\frac{1}{3}}x^4\implies 3^{\frac{3}{3}}x^4\implies 3x^4$