How to Simplify a Square Root
Understand factoring., Divide by the smallest prime number possible., Rewrite the square root as a multiplication problem., Repeat with one of the remaining numbers., Finish simplifying by "pulling out" an integer., Multiply integers together if...
Step-by-Step Guide
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Step 1: Understand factoring.
The goal of simplifying a square root is to rewrite it in a form that is easy to understand and to use in math problems.
Factoring breaks down a large number into two or more smaller factors, for instance turning 9 into 3 x
3.
Once we find these factors, we can rewrite the square root in simpler form, sometimes even turning it into a normal integer.
For example, √9 = √(3x3) =
3.
Follow the steps below to learn this process for more complicated square roots. -
Step 2: Divide by the smallest prime number possible.
If the number under the square root is even, divide it by
2.
If your number is odd, try dividing it by 3 instead.
If neither of these gives you a whole number, move down this list, testing the other primes until you get a whole number result.
You only need to test the prime numbers, since all other numbers have prime numbers as their factors.
For example, you don't need to test 4, because any number divisible by 4 is also divisible by 2, which you already tried. 2 3 5 7 11 13 17 , Keep everything underneath the square root sign, and don't forget to include both factors.
For example, if you're trying to simplify √98, follow the step above to discover that 98 ÷ 2 = 49, so 98 = 2 x
49.
Rewrite the "98" in the original square root using this information: √98 = √(2 x 49). , Before we can simplify the square root, we keep factoring it until we've broken it down into two identical parts.
This makes sense if you think about what a square root means: the term √(2 x 2) means "the number you can multiply with itself to equal 2 x
2." Obviously, this number is 2! With this goal in mind, let's repeat the steps above for our example problem, √(2 x 49): 2 is already factored as low as it will go. (In other words, it's one of those prime numbers on the list above.) We'll ignore this for now and try to divide 49 instead. 49 can't be evenly divided by 2, or by 3, or by
5.
You can test this yourself using a calculator or long division.
Because these don't give us nice, whole number results, we'll ignore them and keep trying. 49 can be evenly divided by seven. 49 ÷ 7 = 7, so 49 = 7 x
7.
Rewrite the problem: √(2 x 49) = √(2 x 7 x 7). , Once you've broken the problem down into two identical factors, you can turn that into a regular integer outside the square root.
Leave all other factors inside the square root.
For example, √(2 x 7 x 7) = √(2)√(7 x 7) = √(2) x 7 = 7√(2).
Even if it's possible to keep factoring, you don't need to once you've found two identical factors.
For example, √(16) = √(4 x 4) =
4.
If we kept on factoring, we'd end up with the same answer but have to do more work: √(16) = √(4 x 4) = √(2 x 2 x 2 x 2) = √(2 x 2)√(2 x 2) = 2 x 2 =
4. , With some large square roots, you can simplify more than once.
If this happens, multiply the integers together to get your final problem.
Here's an example: √180 = √(2 x 90) √180 = √(2 x 2 x 45) √180 = 2√45, but this can still be simplified further. √180 = 2√(3 x 15) √180 = 2√(3 x 3 x 5) √180 = (2)(3√5) √180 = 6√5 , Some square roots are already in simplest form.
If you keep factoring until every term under the square root is a prime number (listed in one of the steps above), and no two are the same, then there's nothing you can do.
You might have been given a trick question! For example, let's try to simplify √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 of these numbers are prime, so they cannot be factored further.
They're all different, so there's no way to "pull out" an integer. √70 cannot be simplified. -
Step 3: Rewrite the square root as a multiplication problem.
-
Step 4: Repeat with one of the remaining numbers.
-
Step 5: Finish simplifying by "pulling out" an integer.
-
Step 6: Multiply integers together if there are more than one.
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Step 7: Write "cannot be simplified" if there are no two identical factors.
Detailed Guide
The goal of simplifying a square root is to rewrite it in a form that is easy to understand and to use in math problems.
Factoring breaks down a large number into two or more smaller factors, for instance turning 9 into 3 x
3.
Once we find these factors, we can rewrite the square root in simpler form, sometimes even turning it into a normal integer.
For example, √9 = √(3x3) =
3.
Follow the steps below to learn this process for more complicated square roots.
If the number under the square root is even, divide it by
2.
If your number is odd, try dividing it by 3 instead.
If neither of these gives you a whole number, move down this list, testing the other primes until you get a whole number result.
You only need to test the prime numbers, since all other numbers have prime numbers as their factors.
For example, you don't need to test 4, because any number divisible by 4 is also divisible by 2, which you already tried. 2 3 5 7 11 13 17 , Keep everything underneath the square root sign, and don't forget to include both factors.
For example, if you're trying to simplify √98, follow the step above to discover that 98 ÷ 2 = 49, so 98 = 2 x
49.
Rewrite the "98" in the original square root using this information: √98 = √(2 x 49). , Before we can simplify the square root, we keep factoring it until we've broken it down into two identical parts.
This makes sense if you think about what a square root means: the term √(2 x 2) means "the number you can multiply with itself to equal 2 x
2." Obviously, this number is 2! With this goal in mind, let's repeat the steps above for our example problem, √(2 x 49): 2 is already factored as low as it will go. (In other words, it's one of those prime numbers on the list above.) We'll ignore this for now and try to divide 49 instead. 49 can't be evenly divided by 2, or by 3, or by
5.
You can test this yourself using a calculator or long division.
Because these don't give us nice, whole number results, we'll ignore them and keep trying. 49 can be evenly divided by seven. 49 ÷ 7 = 7, so 49 = 7 x
7.
Rewrite the problem: √(2 x 49) = √(2 x 7 x 7). , Once you've broken the problem down into two identical factors, you can turn that into a regular integer outside the square root.
Leave all other factors inside the square root.
For example, √(2 x 7 x 7) = √(2)√(7 x 7) = √(2) x 7 = 7√(2).
Even if it's possible to keep factoring, you don't need to once you've found two identical factors.
For example, √(16) = √(4 x 4) =
4.
If we kept on factoring, we'd end up with the same answer but have to do more work: √(16) = √(4 x 4) = √(2 x 2 x 2 x 2) = √(2 x 2)√(2 x 2) = 2 x 2 =
4. , With some large square roots, you can simplify more than once.
If this happens, multiply the integers together to get your final problem.
Here's an example: √180 = √(2 x 90) √180 = √(2 x 2 x 45) √180 = 2√45, but this can still be simplified further. √180 = 2√(3 x 15) √180 = 2√(3 x 3 x 5) √180 = (2)(3√5) √180 = 6√5 , Some square roots are already in simplest form.
If you keep factoring until every term under the square root is a prime number (listed in one of the steps above), and no two are the same, then there's nothing you can do.
You might have been given a trick question! For example, let's try to simplify √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 of these numbers are prime, so they cannot be factored further.
They're all different, so there's no way to "pull out" an integer. √70 cannot be simplified.
About the Author
Brian Diaz
Specializes in breaking down complex lifestyle topics into simple steps.
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