Question

The value of $\sqrt[3]{x^{10}}$, when $x=-2$, can be written in simplest form as $a\sqrt[3]{b}$, where $a=$_______ and $b=$_______.

• A. $8$
• B. $2$

Answer 1

Find the values of $a$ and $b$.

Substitute $-2$ for $x$ in the expression $\sqrt[3]{x^{10}}$to obtain $\sqrt[3]{{(-2)}^{10}}$.

Since ${(-2)}^{10}=1024=2^{10}$. Therefore, the given expression can be written as $\sqrt[3]{2^{10}}$.

The product rule of indices states that $c^m\times c^n=c^{m+n}$, where $c,m$, and $n$ are any real numbers.

Use the product rule of indices to write $2^{10}$ as $2^9\times 2$.

Substitute $2^9\times 2$ for $2^{10}$ in $\sqrt[3]{2^{10}}$ and simplify.

$\sqrt[3]{2^9\times 2}=2^3\sqrt[3]{2}$

$=8\sqrt[3]{2}$

On comparing $8\sqrt[3]{2}$ with the given simplest form $a\sqrt[3]{b}$.

We get, $a=8$ and $b=2$.

Hence, the value of $\sqrt[3]{x^{10}}$, when $x=-2$, can be written in simplest form as $a\sqrt[3]{b}$, where $a=\underline{8}$ and $b=\underline{2}$.