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Square Roots of Products and Fractions


Whenever you need to take the square root of a big number, you can always factorize (rewrite a number as other numbers multiplied together) the number and instead take the square root of its factors, one by one. When you do this, it can be useful to look for ways to factorize your big number into a square number multiplied by something else. In that case, the calculation becomes much simpler. Have a look at the examples in the boxes below.

Rule

The Square Root of a Product

a b = a bfora,b 0

Example 1

Find the square root of 32.

32 = 16 2 = 16 2 = 42

I could have chosen to factorize 32 = 4 8, but taking the square root of those factors wouldn’t help much. 32 = 16 2 is a factorization including a square number, and that makes the calculation much simpler. I leave the answer as 42, because 2 is an exact number while its decimal form will always be an approximation.

Example 2

Find the square root of 50.

50 = 25 2 = 25 2 = 52

Here, you first have to factorize 50 before you take the square root of each factor.

Example 3

Find the square root of 72.

72 = 9 8 = 9 4 2 = 9 4 2 = 3 2 2 = 6 2 You first factorize 72 = 9 4 2. You know the answer to 9 and 4. 2 becomes a never ending decimal number, so you can just leave it as is in the final answer.

Rule

Square root of a fraction

a b = a b

Whenever you want to take the square root a fraction, you can instead take the square root of the numerator and the denominator by themselves.

I’ve said it before and I’ll say it again: You have to memorize the formulas. You won’t stand a chance without them. Be in full control of the square root of a product and a fraction. Here are a couple of examples.

Example 4

Find the square root of 25 16.

25 16 = 25 16 = 5 4

Example 5

Find the square root of 361 169.

361 169 = 361 169 = 19 13

If you want to practice factorization in general, I recommend that you watch instructional videos about factorization.

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How Square Roots Work