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If someone asked you to calculate 75÷3\sqrt{75} \div \sqrt{3} by hand, you might dread finding messy decimals.
Yet with one simple algebraic property, you can combine them and find the exact integer answer in seconds.
How can we pack two separate square roots into a single division problem?

The Quotient Property of Radicals

The radicand is the value inside the radical symbol. The quotient property states that for any non-negative number aa and positive number bb, dividing two square roots equals the square root of their quotient: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
Why does this work? If you square the fraction ab\frac{\sqrt{a}}{\sqrt{b}}, you get (a)2(b)2=ab\frac{(\sqrt{a})^2}{(\sqrt{b})^2} = \frac{a}{b}, which by definition proves that ab\frac{\sqrt{a}}{\sqrt{b}} is the square root of ab\frac{a}{b}.
How do we decide whether to merge two radicals together or keep them apart?

Dividing and Simplifying Radicands

When radicands divide into each other cleanly, combine them under one radical first.
For example, to solve 722\frac{\sqrt{72}}{\sqrt{2}}, rewrite it as 722=36=6\sqrt{\frac{72}{2}} = \sqrt{36} = 6.
If you have coefficients outside the radicals, divide outside numbers with outside numbers, and inside numbers with inside numbers: 125032=(123)502=425=4(5)=20\frac{12\sqrt{50}}{3\sqrt{2}} = \left(\frac{12}{3}\right)\sqrt{\frac{50}{2}} = 4\sqrt{25} = 4(5) = 20