Question

Evaluate $\sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}$ .

Answer 1

Hint: We have to split the powers of 0.5 and 3 using the property ${{a}^{m}}\times {{a}^{n}}={{a}^{mn}}$ so that we will get the result as $\sqrt{{{\left( 0.5 \right)}^{2}}\times 0.5\times 6\times {{3}^{4}}\times 3}$ . Then, we have to use the properties $\sqrt[n]{ab}=\sqrt[n]{a}\times \sqrt[n]{b}$ , $\sqrt[n]{{{a}^{n}}}=a$ and $\sqrt[n]{a}={{\left( a \right)}^{\dfrac{1}{n}}}$ and simplify the result further. Then, we have to write 6 as the product of 2 and 3 and simplify.

Complete step by step answer:
We have to evaluate $\sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}$ . We know that ${{a}^{m}}\times {{a}^{n}}={{a}^{mn}}$ . Therefore, we can write ${{\left( 0.5 \right)}^{3}}$ and ${{3}^{5}}$ in the following manner.
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\sqrt{{{\left( 0.5 \right)}^{2}}\times 0.5\times 6\times {{3}^{4}}\times 3}$
We know that $\sqrt[n]{ab}=\sqrt[n]{a}\times \sqrt[n]{b}$ . Therefore, we can write the above expression as
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\sqrt{{{\left( 0.5 \right)}^{2}}}\times \sqrt{{{3}^{4}}}\times \sqrt{0.5\times 6\times 3}$
We know that $\sqrt[n]{{{a}^{n}}}=a$ , $\sqrt[n]{a}={{\left( a \right)}^{\dfrac{1}{n}}}$ and $\sqrt[n]{{{a}^{m}}}={{a}^{\dfrac{m}{n}}}$ . Therefore, the above equation can be written as
$\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{\dfrac{4}{2}}}\times \sqrt{0.5\times 6\times 3} \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times \sqrt{0.5\times 6\times 3} \\
\end{align}$
Let us write 6 as the product of 2 and 3.
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times \sqrt{0.5\times 2\times 3\times 3}$
We know that ${{a}^{m}}\times {{a}^{n}}={{a}^{mn}}$ . Therefore, the above equation can be written as
$\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times \sqrt{0.5\times 2\times {{3}^{1+1}}} \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times \sqrt{0.5\times 2\times {{3}^{2}}} \\
\end{align}$
We have to apply the rule $\sqrt[n]{ab}=\sqrt[n]{a}\times \sqrt[n]{b}$ .
$\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times \sqrt{{{3}^{2}}}\times \sqrt{0.5\times 2} \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times 3\times \sqrt{0.5\times 2} \\
\end{align}$
Let us multiply 0.5 and 2.
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times 3\times \sqrt{1}$
We know that $\sqrt{1}=1$ .
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times {{3}^{2}}\times 3\times 1$
Let us multiply the terms in the RHS.
$\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=0.5\times 27 \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=13.5 \\
\end{align}$
Hence, the value of $\sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}$ is 13.5.

Note: Students must thoroughly learn and understand the rules of exponents and radicals to solve questions of this type. They must simplify the expression using these rules so that the steps will not be complicated. We can also evaluate $\sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}$ without applying the exponential rules as follows.
We have to write the result of ${{\left( 0.5 \right)}^{2}}\text{ and }{{3}^{5}}$ .
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\sqrt{0.125\times 6\times 243}$
Now, we have to multiply the terms inside the square root.
$\Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\sqrt{182.25}$
Let us multiply and divide the term inside the root by 100.
$\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\sqrt{\dfrac{182.25}{100}\times 100} \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\sqrt{\dfrac{18225}{100}} \\
\end{align}$
We know that $\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$ .
$\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\dfrac{\sqrt{18225}}{\sqrt{100}} \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=\dfrac{135}{10} \\
 & \Rightarrow \sqrt{{{\left( 0.5 \right)}^{3}}\times 6\times {{3}^{5}}}=13.5 \\
\end{align}$