Question

Simplify the following expression:
${\left( {121} \right)^{\dfrac{1}{3}}} \times {\left( {11} \right)^{\dfrac{1}{3}}}$

Answer 1

Hint: We know that 121 is square of 11 i.e. $121 = {11^2}$. Use this to convert the entire expression into the same base. Then use the rules of exponents ${\left( {{a^m}} \right)^n} = {a^{mn}}$ and ${a^m} \times {a^n} = {a^{m + n}}$ to calculate its final value.

Complete step-by-step solution:
According to the question, we have to calculate the value of exponential expression ${\left( {121} \right)^{\dfrac{1}{3}}} \times {\left( {11} \right)^{\dfrac{1}{3}}}$.
Let its value is some variable $x$. Then we have:
$ \Rightarrow x = {\left( {121} \right)^{\dfrac{1}{3}}} \times {\left( {11} \right)^{\dfrac{1}{3}}}$
Now, we know that 121 is the square of 11 as shown below:
$ \Rightarrow 121 = {11^2}$
Using this in the above expression bringing the entire expression in same base, we’ll get:
$ \Rightarrow x = {\left( {{{11}^2}} \right)^{\dfrac{1}{3}}} \times {\left( {11} \right)^{\dfrac{1}{3}}}$
We have a formula of exponents which is:
$ \Rightarrow {\left( {{a^m}} \right)^n} = {a^{mn}}$
Using this formula for the expression, we’ll get:
$ \Rightarrow x = {\left( {11} \right)^{\dfrac{2}{3}}} \times {\left( {11} \right)^{\dfrac{1}{3}}}$
We will use another property of exponents as given below:
$ \Rightarrow {a^m} \times {a^n} = {a^{m + n}}$
Using this property, we’ll get:
$ \Rightarrow x = {\left( {11} \right)^{\dfrac{2}{3} + \dfrac{1}{3}}}$
Simplifying it further, we’ll get:
\[
   \Rightarrow x = {\left( {11} \right)^{\dfrac{{2 + 1}}{3}}} \\
   \Rightarrow x = {\left( {11} \right)^{\dfrac{3}{3}}} = {\left( {11} \right)^1} \\
   \Rightarrow x = 11
 \]

Thus the value of the expression ${\left( {121} \right)^{\dfrac{1}{3}}} \times {\left( {11} \right)^{\dfrac{1}{3}}}$ is 11.

Note: Some of the important properties of exponents are shown below:
$
   \Rightarrow {a^0} = 1 \\
   \Rightarrow {a^m} \times {a^n} = {a^{m + n}} \\
   \Rightarrow \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} \\
   \Rightarrow {\left( {{a^m}} \right)^n} = {a^{mn}}
 $
We use these properties to bring any exponential expression in the same base after which we can calculate the value of the expression easily because we cannot perform any mathematical operations between two terms having different bases.