AI Tutor
If a square garden tile has an area of 2564 sq ft\frac{25}{64}\text{ sq ft}, finding the length of one side means calculating 2564\sqrt{\frac{25}{64}}.
At first glance, taking the square root of a fraction looks messy, but a fundamental property lets you split the top and bottom apart instantly.

The Quotient Property of Square Roots

The radicand is the number or expression sitting under the radical sign, and when your radicand is a fraction, you can take the square root of the numerator and denominator separately.
This rule is the quotient property of radicals, which states that for any non-negative real numbers aa and positive bb:
ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
Why does this work? Because squaring undoes a square root: (ab)2=(a)2(b)2=ab\left(\frac{\sqrt{a}}{\sqrt{b}}\right)^2 = \frac{(\sqrt{a})^2}{(\sqrt{b})^2} = \frac{a}{b}.
In the early 1600s, French mathematician Albert Girard helped formalize fractional exponents, showing that a root is just a power (ab)1/2=a1/2b1/2\left(\frac{a}{b}\right)^{1/2} = \frac{a^{1/2}}{b^{1/2}}, which obeys the exact same exponent laws you already use.
What happens when both the numerator and denominator are perfect squares?

Example 1: Perfect Squares in Both Parts

Let's simplify 4981\sqrt{\frac{49}{81}}.
First, split the expression into two separate radicals: 4981\frac{\sqrt{49}}{\sqrt{81}}.