Question

How do you simplify $\sqrt[3]{125{{x}^{21}}{{y}^{24}}}$ ?

Answer 1

Hint: Here in this question we have to find the cube root of the given expression inside the radical symbol. So we will first try to simplify the expression and then find its cube root. We can express 125 as ${{5}^{3}}$ .

Complete step by step answer:
We have been given an expression $\sqrt[3]{125{{x}^{21}}{{y}^{24}}}$.
We have to simplify the given expression.
Now, in the given expression we have to find the cube root of the expression.
Now, we know that we can write the number 125 as ${{5}^{3}}$ because 125 is the cube of 5.
Now, let us try to express x and y in terms of power 3 then we will get
$\Rightarrow \sqrt[3]{{{5}^{3}}{{x}^{3\times 7}}{{y}^{3\times 8}}}$
Now, again simplifying the above obtained expression we will get
$\Rightarrow \sqrt[3]{{{\left( 5{{x}^{7}}{{y}^{8}} \right)}^{3}}}$
Now, we know that $\sqrt[3]{a}={{\left( a \right)}^{\dfrac{1}{3}}}$
Now, substituting the values we will get
$\Rightarrow {{\left( {{\left( 5{{x}^{7}}{{y}^{8}} \right)}^{3}} \right)}^{\dfrac{1}{3}}}$
Now, we know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
Now, applying the above property to the obtained equation we will get
$\Rightarrow {{\left( 5{{x}^{7}}{{y}^{8}} \right)}^{3\times }}^{\dfrac{1}{3}}$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow {{\left( 5{{x}^{7}}{{y}^{8}} \right)}^{1}} \\
 & \Rightarrow 5{{x}^{7}}{{y}^{8}} \\
\end{align}$
So we get the cube root of the given expression as $5{{x}^{7}}{{y}^{8}}$ .
Hence above is the required simplified form of the given expression.

Note: The simple radical symbol $\sqrt{{}}$ without any number represents the square root and the radical symbol with power three represents the cube root of an expression inside the radical symbol. So be careful while solving such expressions, first understand the meaning of the symbol and then solve further.