Square root simplification

Question

How do you simplify the square root of 648?

Answer 1

First find a large factor, like 81. 81 goes into 648 8 times. Root 81 times root 8 gets root 648. Break up the eight into root 4 and root 2. You are essentially trying to pull as many numbers out of the root sign as possible. The simplification would be 9 * 2 * root 2 or 18 root 2.

Answer 2

To simplify the square root of 648, you need to find its factors. Keep in mind that when you find the factors, being able to find the square root of them is helpful. What we're trying to do is simplify the number 648 into factors so it's easier to simplify. Let's look at all of the factors of 648:

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648.

To get the square root of 648, I know that one of the factors is 81 because we can do the square root of 81. So, we have our list of factors, but an easy way to know what the other factor is to do 648 ÷ 81 = 8.

So, 81 x 8 = 648. To break it down even more and make it easier, you can write it in exponential form

2³ x 3⁴ = 648

Now, we can do the square root of 648. So, √648 = √2³ x 3⁴ . The square of √2³ isn't possible to get an exact number. But, we can break down 2³ even more to simplify. 2³ = 2² x 2 because we are following the product rule (a^m x aⁿ = a^{m + n}) The square root of 2² is 2, so all we have left is 2 x √2. The square root of 3⁴ is 9 because √81 = 9

So, we have 2 x 9 x √2 = 18√2

The square root of 648 is 18√2.

To get the square root of a number in its simplest form, follow these three steps:

Step 1 - Find the factors

Step 2 - Find the square roots of each of those factors

Step 3 - After finding the square root of each factor, combine them and finish simplifying