Adding & subtracting square roots

Japanese school year: Junior high 3

What you learn

Learn how to add and subtract radical expressions by treating identical radicals like combined algebraic terms. This method is regularly used when calculating geometric lengths and simplifying solutions to quadratic equations. Reviewing how to simplify square roots and combine like terms beforehand is essential for quick mastery.

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Key points

Use this to add radical terms sharing the same radicand $a$ . Just like combining like terms in algebra ( $3x + 2x = 5x$ ), add the coefficients $m$ and $n$ in front of $\sqrt{a}$ .

m\sqrt{a} + n\sqrt{a} = (m + n)\sqrt{a} \quad (a > 0)

Use this to subtract radical terms sharing the same radicand $a$ . Just like addition, subtract the coefficient $n$ from $m$ while retaining $\sqrt{a}$ .

m\sqrt{a} - n\sqrt{a} = (m - n)\sqrt{a} \quad (a > 0)

This is the preparation step before adding or subtracting square roots. First simplify each term into $a\sqrt{b}$ form so the radicands are minimal, then combine like terms.

A common misconception in radical addition: the sum $\sqrt{a} + \sqrt{b}$ cannot be combined into a single radical $\sqrt{a + b}$ .

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Practice 1Coming soon