Solve Division of Square Roots: √36/√9 Step-by-Step

Solve the following exercise:

$\frac{\sqrt{36}}{\sqrt{9}}=$

Express the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

$$Remember that for a square root (also called "root to the power of 2") we don't write the root's power:

$n=2$

meaning:

$\sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}$

Thus we will proceed to convert all the roots in the problem to powers:

$\frac{\sqrt{36}}{\sqrt{9}}=\frac{36^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Below is the power law for a fraction inside of parentheses:

$\frac{a^n}{c^n}= \big(\frac{a}{c}\big)^n$

However in the opposite direction,

Note that both the numerator and denominator in the last expression that we obtained are raised to the same power. Which means that we can write the expression using the above power law as a fraction inside of parentheses and raised to a power:
$\frac{36^{\frac{1}{2}}}{9^{\frac{1}{2}}}=\big(\frac{36}{9}\big)^{\frac{1}{2}}$

We can only do this because both the numerator and denominator of the fraction were raised to the same power,

Let's summarize the different steps of our solution so far:

$\frac{\sqrt{36}}{\sqrt{9}}=\frac{36^{\frac{1}{2}}}{9^{\frac{1}{2}}} =\big(\frac{36}{9}\big)^{\frac{1}{2}}$

Proceed to calculate (by reducing the fraction) the expression inside of the parentheses:

$\big(\frac{36}{9}\big)^{\frac{1}{2}} =4^\frac{1}{2}$

and we'll return to the root form using the definition of root as a power mentioned above, ( however this time in the opposite direction):

$$$$$a^{\frac{1}{n}}=\sqrt[n]{a}$

Let's apply this definition to the expression that we obtained:

$4^\frac{1}{2}=\sqrt[2]{4}\ =\sqrt{4}=2$

Once in the last step we calculate the numerical value of the root of 4,

To summarize we obtained the following calculation: :

$\frac{\sqrt{36}}{\sqrt{9}}=\big(\frac{36}{9}\big)^{\frac{1}{2}} =\sqrt{4}=2$

Therefore the correct answer is answer B.