Question

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

Answer 1

`\bf ~\hspace{7em}\textit{rational exponents} \\\\ a^{( n)/( m)} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-( n)/( m)} \implies \cfrac{1}{a^{( 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)^{(1)/(3)}\cdot (3x^8)^{(1)/(3)}\implies 9^{(1)/(3)}\cdot x^{4\cdot (1)/(3)}\cdot 3^{(1)/(3)}\cdot x^{8\cdot (1)/(3)}`

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

Answer 2

`3x^4`

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`