3 Ways to Simplify Square Roots

2024 Author: Jason Gerald | [email protected]. Last modified: 2023-12-16 10:50

Simplifying the square root isn't as difficult as it seems. To simplify the square root, you just have to factor the number and take the square root of whatever perfect square is below the square root. If you remember commonly used perfect squares and know how to factor numbers, you'll be able to simplify square roots pretty well.

Step

Method 1 of 3: Simplifying Square Roots by Factoring

Step 1. Understand about factors

The goal of simplifying square roots is to write them in a form that is easy to understand and use in math problems. By factoring, a large number is broken down into two or more smaller "factor" numbers, for example changing 9 to 3 x 3. Once we find this factor, we can rewrite the square root in a simpler form, sometimes even changing it be a regular integer. For example, 9 = (3x3) = 3. Follow these steps to learn about this process in more complex square roots.

Step 2. Divide the number by the smallest possible prime number

If the number under the square root is an even number, divide by 2. If your number is odd, then try dividing by 5. If neither of these divisions gives you an integer, try the next number in the list below, dividing by each number. prime to get an integer as the result. You only need to test for prime numbers, because all other numbers have prime numbers as factors. For example, you don't need to test with the number 4, because all numbers that are divisible by 4 are also divisible by 2, which you have tried before.

Step 3. Rewrite the square root as a multiplication problem

Keep writing this multiplication under the square root, and don't forget to include both factors. For example, if you are trying to simplify 98, follow the steps above to find that 98 2 = 49, so 98 = 2 x 49. Rewrite the number "98" in its original square root using this information: 98 = (2 x 49).

Step 4. Repeat on one of the remaining numbers

Before we can simplify the square root, we need to keep factoring it until it becomes two exactly equal numbers. This makes sense if you remember what the square root means: the number (2 x 2) means "a number that you can multiply by itself equals 2 x 2." Of course, the answer is 2! With this in mind, let's repeat the steps above to solve our example problem (2 x 49):

2 has been factored as small as possible. (In other words, this number is one of the prime numbers listed above). We'll ignore this number for now and try dividing by 49 first.
49 can't be completely divided by 2, or by 3, or by 5. You can test this yourself using a calculator or using long division. Since this division doesn't give a whole number, we'll ignore it and try the next number.
49 is completely divisible by 7. 49 7 = 7, so 49 = 7 x 7.
Rewrite the problem above with: (2 x 49) = (2 x 7 x 7).

Step 5. Solve by "extracting" an integer

Once you've solved the problem into two exactly equal factors, you can convert it to a regular integer outside of the square root. Let the rest of the factors remain in the square root. For example, (2 x 7 x 7) = (2)√(7 x 7) = (2) x 7 = 7√(2).

Even if you can still factor further, you don't have to do it again once you find two factors that match exactly. For example, (16) = (4 x 4) = 4. If we keep factoring, we will get the same answer but in a longer way: (16) = (4 x 4) = (2 x 2 x 2 x 2) = (2 x 2)√(2 x 2) = 2 x 2 = 4

Step 6. Multiply all integers if there is more than one

For some large square roots, you can simplify more than once. If this is the case, multiply the integer you get to get the final answer. Here's an example:

180 = (2 x 90)
180 = (2 x 2 x 45)
180 = 2√45, but this value can be simplified further.
180 = 2√(3 x 15)
180 = 2√(3 x 3 x 5)
√180 = (2)(3√5)
√180 = 6√5

Step 7. Write down "cannot be simplified" if no two factors are equal

Some square root numbers are already in their simplest form. If you keep factoring until all of them are prime numbers (as listed in the step above), and none of the pairs are the same, then there's nothing you can do. You might be given a trap question! For example, try simplifying 70:

70 = 35 x 2, so 70 = (35 x 2)
35 = 7 x 5, so (35 x 2) = (7 x 5 x 2)
All three numbers here are prime numbers, so they can't be factored any further. The three numbers are different, so it is impossible to produce an integer. 70 cannot be simplified.

Method 2 of 3: Recognizing Perfect Squares

Step 1. Remember some perfect squares

Squaring a number, or multiplying it by the number itself, creates a perfect square. For example, 25 is a perfect square, because 5 x 5, or 5², equals 25. Remember at least the first ten perfect squares to help you identify and simplify perfect square roots. Here are the first ten perfect square numbers:

1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100

Step 2. Find the square root of the perfect square

If you recognize a perfect square under the square root, you can immediately convert it to a square root and remove it from the sign (√). For example, if you see the number 25 under the square root, you already know the answer is 5, because 25 is a perfect square. This list is the same as above, starting from the square root to the answer:

√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√64 = 8
√81 = 9
√100 = 10

Step 3. Factor the number into a perfect square

Take advantage of perfect squares when continuing with the factor method of simplifying square roots. If you are aware of the factors of a perfect square, then you will be faster and easier to solve problems. Here are some tips you can use:

50 = (25 x 2) = 5√2. If the last two digits of a number end in 25, 50, or 75, you can always factor 25 of that number.
1700 = (100 x 17) = 10√17. If the last two numbers end in 00, then you can always factor 100 of that number.
72 = (9 x 8) = 3√8. Get to know the multiplication of nine to make it easier for you. Here's a tip for recognizing them: if "all" of the numbers in a number add up to nine, then nine is a factor.
12 = (4 x 3) = 2√3. No specific tips here, but it's usually easy to check if a small number is divisible by 4. Keep this in mind when looking for other factors.

Step 4. Factor a number with more than one perfect square

If the factors of a number have more than one perfect square, take them all out of the square root. If you get multiple perfect squares in the process of simplifying the square root, move all the square roots outside the sign and multiply them all together. For example, try to simplify 72:

72 = (9 x 8)
72 = (9 x 4 x 2)
72 = (9) x (4) x (2)
72 = 3 x 2 x 2
√72 = 6√2

Method 3 of 3: Understanding the Terms

Step 1. Know that the square root sign (√) is the square root sign

For example, in problem 25, "√" is the root sign.

Step 2. Know the radicand is the number inside the root sign

This is the number you have to calculate the square root of. For example, in the problem of 25, "25" is the square root.

Step 3. Know that the coefficient is a number outside the square root

This number is the square root of the multiplier; this number is to the left of the root sign. For example, in problem 7√2, "7" is the value of the coefficient.

Step 4. Know that a factor is a number that is fully divisible by a number

For example, 2 is a factor of 8 because 8 4 = 2, but 3 is not a factor of 8 because 8÷3 doesn't give a whole number. Just like in the other examples, 5 is a factor of 25 because 5 x 5 = 25.

Step 5. Understand the meaning of simplification of the square root

Simplifying the square root simply means factoring the perfect square of the square root, removing it to the left of the radical sign, and leaving the remaining factors below the radical sign. If a number is a perfect square then the square root will disappear when you write down the root. For example, 98 can be simplified to 7√2.

Tips

One way to find a perfect square that can be factored into a number is to look at a list of perfect squares, starting with the lesser than your square root, or with the number below the square root. For example, when looking for a perfect square that is not greater than 27, start with 25 and work your way down to 16 and "stop at 9", when you find a perfect square that divides 27

Warning

Simplifying is not the same as calculating the value. None of the steps in this process require you to get a number with a decimal in it.
Calculators can be helpful for large numbers, but the more you practice on your own, the easier it will be to simplify square roots.