Question

What is 4 square root 5 squared?

Answer 1

Hint: First of all form the mathematical expression using the provided information. To form the mathematical expression, take the square root of 5 and multiply it with 4 to get the radical expression. Now, square this expression obtained and use the formula of exponents given as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to square the radical term and get the answer.

Complete step by step solution:
Here we have been asked to find the value of the mathematical expression that will be formed using the sentence ‘4 square root 5 squared’. Let us assume the required expression as E.
Now, according to the given sentence we have to take the square root of 5 and multiply it with 4, therefore the expression E can be written as: -
$\begin{align}
  & \Rightarrow E=4\times \sqrt{5} \\
 & \Rightarrow E=4\sqrt{5} \\
\end{align}$
Further, we need to square this expression, so on squaring both the sides we get,
$\Rightarrow {{E}^{2}}={{\left( 4\sqrt{5} \right)}^{2}}$
Using the formula of exponents given as ${{\left( a\times b \right)}^{n}}={{a}^{n}}\times {{b}^{n}}$ we get,
$\begin{align}
  & \Rightarrow {{E}^{2}}={{\left( 4 \right)}^{2}}\times {{\left( \sqrt{5} \right)}^{2}} \\
 & \Rightarrow {{E}^{2}}=16\times {{\left( \sqrt{5} \right)}^{2}} \\
\end{align}$
The radical expression $\sqrt{5}$ can be written in the exponential form as ${{\left( 5 \right)}^{\dfrac{1}{2}}}$ because we have the square root form of the radical. So we get,
$\Rightarrow {{E}^{2}}=16\times {{\left( {{5}^{\dfrac{1}{2}}} \right)}^{2}}$
Using the formula of exponents given as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ we get,
\[\begin{align}
  & \Rightarrow {{E}^{2}}=16\times \left( {{5}^{\dfrac{1}{2}\times 2}} \right) \\
 & \Rightarrow {{E}^{2}}=16\times 5 \\
 & \therefore {{E}^{2}}=80 \\
\end{align}\]
Hence, the value of the given sentence is 80.

Note: You must be careful while forming the mathematical expression according to the words given in the question because one wrong operation can make all the steps of the solution wrong. You must remember all the formulas of the topic ‘exponents and powers’ as they are used in several chapters of mathematics. Some basic formulas of exponents are: - ${{a}^{m}}\times {{a}^{m}}={{a}^{m+n}}$, ${{a}^{m}}\div {{a}^{m}}={{a}^{m-n}}$, ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$, ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ etc.