Q:

# How do you simplify square roots?

A:

Square roots, designated by the √ symbol, are operations that find a number which, when multiplied by itself, create the number under the radical symbol. Simplifying the square root of a number, or radicand, requires factoring and distribution.

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1. Factor the radicand

Use prime factorization to determine the parts that make up the number under the √. For example, the number 360 can be factored into the prime numbers 2, 2, 2, 3, 3 and 5, because 2*2*2*3*3*5 = 360.

2. Find squares in the factors

A number multiplied by itself is that number squared. Therefore, any pairs of the same number in the prime factorization are squares. For 360, or 2*2*2*3*3*5, both the 2s and the 3s have pairs. 2*2=4, and 3*3=9, both of which have whole-number square roots. A perfect square multiplied by another perfect square creates a product that is also a perfect square. For example, 4*9=36, and √36=6.

3. Use distribution

Multiply the perfect squares together, and then multiply the other prime factors together. Distribute the square root to each value because √(a*b) = √(a) * √(b). √360 = √(2*2*2*3*3*5) = √((2*2*3*3)*(2*5)) = √(36*10) = √(36) * √(10) Take the square roots of any perfect squares. Leave the other prime factors in radical form. √360 = √(36) * √(10) = 6 * √(10) = 6√10

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