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

Let's use the definition of root as a power:

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

$$when we remember that in a square root (also called "root to the power of 2") we don't write the root's power and:

$n=2$

meaning:

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

We will convert therefore all the roots in the problem to powers:

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

Now let's recall the power law for a fraction in parentheses:

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

But in the opposite direction,

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

We emphasize that we could do this only because both the numerator and denominator of the fraction were raised to the same power,

Let's summarize our solution steps so far we got that:

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

Now let's calculate (by reducing the fraction) the expression inside 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, but in the opposite direction:

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

Let's apply this definition to the expression we got:

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

where in the last step we calculated the numerical value of the root of 4,

Let's summarize the solution steps, we got that:

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

Therefore the correct answer is answer B.