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Simplifying radicals

Square roots

Japanese school year: Junior high 3

What you learn

Learn how to simplify radical expressions by taking square factors out of the radical into their simplest radical form. This skill is indispensable for performing arithmetic with radicals and expressing solutions cleanly. Prior understanding of prime factorization and the definition of square roots will ensure a smooth learning experience.

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Key points

Use this to simplify the number inside a radical to the smallest possible integer. If there is a squared factor a2a^2 inside, take aa outside the square root.

a2b=ab(a>0,  b>0)\sqrt{a^2b} = a\sqrt{b} \quad (a > 0, \; b > 0)

Use this when the denominator inside a radical is a perfect square, b2b^2. Taking bb outside the root into the denominator leaves the square root only in the numerator.

ab2=ab(a>0,  b>0)\sqrt{\frac{a}{b^2}} = \frac{\sqrt{a}}{b} \quad (a > 0, \; b > 0)

Use this when finding the principal square root of a squared term. Since principal roots are always non-negative, if aa is negative, it evaluates to its absolute value ∣a∣|a|.

a2=∣a∣\sqrt{a^2} = |a|

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