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Rationalizing denominators

Square roots

Japanese school year: Junior high 3

What you learn

You will learn how to eliminate radicals from the denominator of a fraction to simplify expressions. This technique makes calculating numerical values and comparing expressions much easier. A good understanding of square root properties and algebraic expansion is helpful before starting this topic.

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Key points

Use this formula to eliminate the square root from the denominator of a fraction. Multiplying both the numerator and the denominator by a\sqrt{a} turns the denominator into a rational number aa.

ca=caa\frac{c}{\sqrt{a}} = \frac{c\sqrt{a}}{a}

Use this formula when the denominator is the sum of two square roots. Multiplying both the numerator and the denominator by (a−b)(\sqrt{a} - \sqrt{b}) clears the square roots from the denominator.

ca+b=c(a−b)a−b\frac{c}{\sqrt{a} + \sqrt{b}} = \frac{c(\sqrt{a} - \sqrt{b})}{a - b}

Use this formula when the denominator consists of a regular number aa and a square root b\sqrt{b}. Multiplying both top and bottom by (a−b)(a - \sqrt{b}) eliminates the root, giving a2−ba^2 - b in the denominator.

ca+b=c(a−b)a2−b\frac{c}{a + \sqrt{b}} = \frac{c(a - \sqrt{b})}{a^2 - b}

Choose a set to practice.