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Nested radicals

Square roots

Japanese school year: Math I

What you learn

Learn how to transform nested radicals into the form of √a ± √b, expressing them as sums or differences of square roots. This technique simplifies expressions involving radicals and makes irrational values easier to handle. Prior mastery of algebraic expansions and basic square root arithmetic supports smooth learning.

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Key points

This formula simplifies a nested square root. When the inner root has a coefficient of 2, find two positive numbers aa and bb that add up to a+ba + b and multiply to abab.

(a+b)+2ab=a+b(a>0,b>0)\sqrt{(a + b) + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b} \quad (a > 0, b > 0)

Use this formula to simplify a nested root with subtraction. Since the square root is never negative, always choose a>ba > b and subtract the smaller b\sqrt{b} from the larger a\sqrt{a}.

(a+b)−2ab=a−b(a>b>0)\sqrt{(a + b) - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b} \quad (a > b > 0)

This technique is used when the coefficient in front of the inner root is not 2. Move factors into or out of the radical to produce a 2, then apply the formula.

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