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XVIII Square Root Stuff

Breaking Down Square Roots of Large Numbers

Break the number into its factors and multiply the square roots of those.
202=4×52=252\sqrt[2]{20} = \sqrt[2]{4 \times 5} = 2\sqrt[2]{5}

Fractions

General rule: ab2=a2b2\sqrt[2]{\frac a b} = \frac{\sqrt[2]{a}}{\sqrt[2]{b}}

e.g. 9102=92102=3102\sqrt[2]{\frac{9}{10}} = \frac{\sqrt[2]{9}}{\sqrt[2]{10}} = \frac{3}{\sqrt[2]{10}}

However solutions never contain square roots in a denominator. Therefore the denominator has to be rationalized. Since 102102=1\frac{\sqrt[2]{10}}{\sqrt[2]{10}} = 1, this process is as simple as multiplying the result with a fraction that contains its denominator on both sides:

3102×102102=310210\frac{3}{\sqrt[2]{10}} \times \frac{\sqrt[2]{10}}{\sqrt[2]{10}} = \frac{3\sqrt[2]{10}}{10}

This works in general because x2×x2=x\sqrt[2]{x} \times \sqrt[2]{x} = x

Also fractions should be simplified where possible e.g. 4022=202\sqrt[2]{\frac{40}{2}} = \sqrt[2]{20}

Square Root Term in Polynomial Denominator

Given a fraction such as 22−1\frac {2}{\sqrt{2}-1}, forming a difference of squares gets rid of the root in the denominator:

a2−b2=(a+b)(a−b)a^2-b^2=(a+b)(a-b)

So

2(2+1)2−1\frac {2(\sqrt{2}+1)}{2-1}

Misc Notes

x22=x\sqrt[2]{x^2} = x