Question

How would you simplify the square root of $\left( \dfrac{3}{8} \right)$ ?

Answer 1

Hint: We know the property of exponential function $\dfrac{{{a}^{x}}}{{{b}^{x}}}$ is equal to ${{\left( \dfrac{a}{b} \right)}^{x}}$ and ${{\left( ab \right)}^{x}}$ is equal to ${{a}^{x}}{{b}^{x}}$ where the value of b is not equal to 0 . Square root of any number y means y to the power $\dfrac{1}{2}$ , so we can apply the exponential property where power is equal to $\dfrac{1}{2}$

Complete step by step answer:
We have to find the value of square root of $\left( \dfrac{3}{8} \right)$
We know that square root of $\left( \dfrac{3}{8} \right)$ means ${{\left( \dfrac{3}{8} \right)}^{\dfrac{1}{2}}}$
We know the formula ${{\left( \dfrac{a}{b} \right)}^{x}}$ is equal to $\dfrac{{{a}^{x}}}{{{b}^{x}}}$ . we can apply this ${{\left( \dfrac{3}{8} \right)}^{\dfrac{1}{2}}}$
So ${{\left( \dfrac{3}{8} \right)}^{\dfrac{1}{2}}}$ is equal to $\dfrac{{{3}^{\dfrac{1}{2}}}}{{{8}^{\dfrac{1}{2}}}}$
We can not further solve $\sqrt{3}$ , it is an irrational number so we can write an approximate value , we can not write the exact value so let’s not change it.
We can write $\sqrt{8}$ as $\sqrt{4\times 2}$ , by applying the formula ${{\left( ab \right)}^{x}}$ is equal to ${{a}^{x}}{{b}^{x}}$ we can write
$\sqrt{8}=\sqrt{4}\times \sqrt{2}$
 The value of $\sqrt{4}$ is equal to 2
So $\sqrt{8}=2\sqrt{2}$
In $\dfrac{\sqrt{3}}{\sqrt{8}}$ we can multiply $\sqrt{8}$ in both numerator and denominator
$\Rightarrow \dfrac{\sqrt{3}}{\sqrt{8}}=\dfrac{\sqrt{8}\times \sqrt{3}}{8}$
Now we can replace $\sqrt{8}$ by $2\sqrt{2}$
$\Rightarrow \dfrac{\sqrt{3}}{\sqrt{8}}=\dfrac{2\sqrt{2}\times \sqrt{3}}{8}$
Now we can cancel out 2 in numerator and denominator and multiplication of $\sqrt{2}$ and $\sqrt{3}$ will be $\sqrt{6}$
$\Rightarrow \dfrac{\sqrt{3}}{\sqrt{8}}=\dfrac{\sqrt{6}}{4}$
So the value of square root of $\left( \dfrac{3}{8} \right)$ is equal to $\dfrac{\sqrt{6}}{4}$

Note: We can check whether our answer is correct or not by cross multiplication, the value of $4\sqrt{3}$ is equal to $\sqrt{48}$, so our answer is correct. Always remember the standard exponential formula like $\dfrac{{{a}^{x}}}{{{b}^{x}}}$= ${{\left( \dfrac{a}{b} \right)}^{x}}$ , ${{\left( ab \right)}^{x}}$ = ${{a}^{x}}{{b}^{x}}$ . In the formula $\dfrac{{{a}^{x}}}{{{b}^{x}}}$= ${{\left( \dfrac{a}{b} \right)}^{x}}$ keep in mind that b is not equal to 0, we can not make denominator of any fraction 0.