Question

What is the square root of 3 times the square root of 3?

Answer 1

Hint: First of all we will write the required mathematical expression of the given sentence. Now, we will write the square root expression in exponential form. In the next step we are going to use the formula of exponents given as ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ and simplify the exponent to get the answer.

Complete step by step answer:
Here we have been asked to find the value of the expression which will be formed with the given sentence ‘the square root of 3 times the square root of 3’. Here we will be required to use the basic formulas of the topic ‘exponents and powers’. Let us assume the expression that we will form as ‘E’.
Now, according to the sentence given in the question we have to take two numbers and both of them will be square of 3 and we need to multiply them because the term ‘times’ means product. Therefore, the mathematical expression will be given as:
$\Rightarrow E=\sqrt{3}\times \sqrt{3}$
The above expression is in the radical form. Let us change it into the exponential form. We know that the term ‘square root’ means exponent equal to half $\left( \dfrac{1}{2} \right)$, so we have,
$\Rightarrow E={{3}^{\dfrac{1}{2}}}\times {{3}^{\dfrac{1}{2}}}$
Using the formula of exponents ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ we get,
$\begin{align}
  & \Rightarrow E={{3}^{\dfrac{1}{2}+\dfrac{1}{2}}} \\
 & \Rightarrow E={{3}^{1}} \\
 & \therefore E=3 \\
\end{align}$

Hence the simplified value of the given expression is 3.

Note: Here you can also write the given obtained expression as \[{{\left( \sqrt{3} \right)}^{2}}\] and then use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify. You must remember all the formulas of exponents like: - \[{{x}^{m}}\times {{x}^{n}}={{x}^{m+n}}\], \[{{x}^{m}}\div {{x}^{n}}={{x}^{m-n}}\], \[{{\left( {{x}^{m}} \right)}^{n}}={{x}^{m\times n}}\], \[{{x}^{-m}}=\dfrac{1}{{{x}^{m}}}\] etc, as they are used in certain other topics of mathematics. Now, if the bases are different and exponent is the same then we can apply these formulas: ${{a}^{m}}\times {{b}^{m}}={{\left( a\times b \right)}^{m}},{{a}^{m}}\div {{b}^{m}}={{\left( a\div b \right)}^{m}}$ to simplify. In the above question do not try to find the decimal form of $\sqrt{3}$ first and then multiply because it is an irrational number whose decimal form will be non – repeating and non – terminating.