Question

How do you find the square root of 3 to the second power ?

Answer 1

Hint:In order to find the solution of this question , we need to first understand the mathematical terms and should frame to make up the expression . The term ‘ square root ‘ signifies that what should be that number multiplied by itself so that it is the number under the square root symbol called radical . Radical is represented by $\sqrt {} $. For example = 4 , the base part of the value inside the square root such that it contains perfect squares in it that is also by the definition when the number 2 multiplied by itself is equal to the number under the radical or square root symbol.

Formula used:
${(a)^{\dfrac{m}{n}}}$=${({a^m})^{\dfrac{1}{n}}}$
${x^{m + n}} = {x^m} \times {x^n}$

Complete step by step answer:
We are given with the mathematical statements which when framed makes up the expression as -
${\left( {\sqrt 3 } \right)^2}$
First we will try to simplify further by the definition that what should be that number multiplied by itself so that it is the number under the square root symbol . But the value $\sqrt 3 $ cannot be simplified into its factors, So the root we cannot simplify further comes to be surd . Now let’s find the answer by using some formula
\[{({b^m})^n} = {b^{mn}}\]
Also , we know that the square root have the power as $\dfrac{1}{2}$
\[\surd b = {b^{\dfrac{1}{2}}}\]
Some rules of the exponents when applied , we got
$\sqrt 3 = {3^{\dfrac{1}{2}}}$
Let’s consider the formula concerning exponents ${(a)^{\dfrac{m}{n}}}$=${({a^m})^{\dfrac{1}{n}}}$, we get
${\left( {\sqrt 3 } \right)^2}$
Rewriting square root as half power ,
${\left( {{3^{\dfrac{1}{2}}}} \right)^2}$
 By applying ,
\[{({b^m})^n} = {b^{mn}}\]
\[\Rightarrow {3^{\dfrac{1}{2} \times 2}}\]
2 gets canceled out and from the above we can say that ${3^1} = 3$.

Therefore, ${\left( {\sqrt 3 } \right)^2}$ in the simplest form $3$.

Note:Do not Forget to cross check the answer.
-The value of root 3 is approximately equal to $1.732$.
-Also root 3 can also be said as a irrational number as it cannot be expressed in the form of a fraction .
-Under the radical symbol $\sqrt {} $ we cannot place a negative number , it should be the positive real number .
-The two halves can't make up the whole one in the same way the two square roots ( having powers as $\dfrac{1}{2}$) can make up the whole one.For example = $\sqrt 3 $multiplied by $\sqrt 3 $gives the number 3 as the base is same so power gets added which makes the power 1 giving us the real positive number 3 .