Question

What is the following product? 3√16x^7 3√12x^9

Please help, Thank you!

Answer 1

Product of cube root of :

`\sqrt[3]{16x^7} * \sqrt[3]{12x^9}`

First simplify :
`\sqrt[3]{16x^7}`

Factor: One of two or more expressions that are multiplied together to get a product

then, we can write it as:

`\sqrt[3]{2 \cdot 8 x^7}` ,

Rewrite 8 as
`2^3` and
`x^7 = x^6 \cdot x`

`\sqrt[3]{2\cdot 2^3 \cdot x^6 \cdot x}` or

`\sqrt[3]{2\cdot 2^3 \cdot (x^2)^3 \cdot x}` [∵
`(a^x)^y=a^(xy)` ]

or
`\sqrt[3]{2^3 \cdot (x^2)^3 \cdot 2\cdot x}`

or
`2 \cdot x^2\sqrt[3]{2\cdot x}` [∵
`\sqrt[3]{a^3} =a` ]

Similarly, we simplify for
`\sqrt[3]{12 x^9}`

Then, we can write it as
`\sqrt[3]{12\cdot (x^3)^3}` or

`x^3 \cdot \sqrt[3]{12}`

Use :
`x^(a+b)=x^a \cdot x^b` ,
`\sqrt[3]{a} \cdot\sqrt[3]{b} = \sqrt[3]{a \cdot b}`

Now,

`\sqrt[3]{16x^7} * \sqrt[3]{12x^9}` =
`2 \cdot x^2\sqrt[3]{2\cdot x} * x^3 \cdot \sqrt[3]{12}`

=
`2x^2 \cdot x^3 \sqrt[3]{2x} \cdot \sqrt[3]{12}`

=
`2 x^5\cdot \sqrt[3]{2x \cdot 12}` or
`2x^5 \cdot \sqrt[3]{24 x}`

=
`2x^5 \cdot \sqrt[3]{8 \cdot 3x}` or
`2 x^5 \cdot \sqrt[3]{2^3 \cdot 3 \cdot x}`

=
`2x^5 \cdot 2 \sqrt[3]{3x}` =
`4 x^5 \cdot \sqrt[3]{3x}`

therefore, the product of
`\sqrt[3]{16x^7} * \sqrt[3]{12x^9}` is,
`4 x^5 \cdot \sqrt[3]{3x}`

Answer 2

`\displaystyle\sqrt[3]{16x^7}*\sqrt[3]{12x^9}=\sqrt[3]{16\cdot 12x^((7+9))}\\\\=\sqrt[3]{4^3\cdot 3x^(16)}=\sqrt[3]{\left(4x^5\right)^3\cdot 3x}\\\\=4x^5\sqrt[3]{3x}`