Question

What is the following product? 3√16x^7 3√12x^9 Please help, Thank you!

Answer 1

Answer:

Product of cube root of :

$\sqrt[3]{16x^7} \times \sqrt[3]{12x^9}$

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,

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

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

= $2 x^5\cdot \sqrt[3]{2x \cdot 12}$ 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  $\sqrt[3]{16x^7} \times \sqrt[3]{12x^9}$ is,  $4 x^5 \cdot \sqrt[3]{3x}$

Answer 2

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