Page:The Algebra of Mohammed Ben Musa (1831).djvu/44

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that you multiply it by two) then it will suffice to multiply two by two, and then by the square;^[1] the root of the product is equal to twice the root of the original square.

If you require to take it thrice, you multiply three by three, and then by the square; the root of the product is thrice the root of the original square.

Compute in this manner every multiplication of the roots, whether the multiplication be more or less than two.^[2]

(20) If you require to find the moiety of the root of the square, you need only multiply a half by a half, which is a quarter; and then this by the square: the root of the product will be half the root of the first square.^[3]

Follow the same rule when you seek for a third, or a quarter of a root, or any larger or smaller quota^[4] of it, whatever may be the denominator or the numerator.

Examples of this: If you require to double the root of nine,^[5] you multiply two by two, and then by nine: this gives thirty-six; take the root of this, it is six, and this is double the root of nine.

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1. $2{\sqrt {x^{2}}}={\sqrt {4x^{2}}}$
   $3{\sqrt {x^{2}}}={\sqrt {9x^{2}}}$
2. $n{\sqrt {x^{2}}}={\sqrt {n^{2}x^{2}}}$
3. ${\tfrac {1}{2}}{\sqrt {x^{2}}}={\sqrt {\tfrac {x^{2}}{4}}}$
4. ${\tfrac {1}{n}}{\sqrt {x^{2}}}={\sqrt {\tfrac {x^{2}}{n^{2}}}}$
5. $2{\sqrt {9}}={\sqrt {4\times 9}}={\sqrt {36}}=6$