3 Ways to Simplify a Square Root

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3 Ways to Simplify a Square Root
3 Ways to Simplify a Square Root

Video: 3 Ways to Simplify a Square Root

Video: 3 Ways to Simplify a Square Root
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Simplifying a square root is not as difficult as it sounds. To do this, you just factor the number and take the roots of any perfect squares you find. Once you've memorized some common perfect squares and know how to factor a number, you're well on your way to simplifying a square root.

Steps

Method 1 of 3: Simplifying a Square Root Through Factorization

Simplify a Square Root Step 1
Simplify a Square Root Step 1

Step 1. Understand factorization

The purpose of simplifying a square root is to rewrite it in a form that is simple to understand and use in math problems. Factoring breaks a large number into two or more smaller factors, for example, turning 9 into 3 x 3. Once we find these factors, we can rewrite the square root in a simpler form, sometimes even turning it into a normal integer. For example, √9 = √(3x3) = 3. Follow the steps below to learn how to do this process with more complicated square roots.

Simplify a Square Root Step 2
Simplify a Square Root Step 2

Step 2. Divide by the smallest possible prime number

If the number below the square root is even, divide it by 2. If it's odd, try dividing it by 3 instead. If none of these give you a whole number, go through this list testing the other primes until you get a whole number as a result. You only need to test the primes, as all others have prime factors. For example, you don't need to test 4, as any number divisible by 4 is also divisible by 2, which you've already tried.

  • 2.
  • 3.
  • 5.
  • 7.
  • 11.
  • 13.
  • 17.
Simplify a Square Root Step 3
Simplify a Square Root Step 3

Step 3. Rewrite the square root as a multiplication problem

Leave everything under the root and be sure to include both factors. For example, if you are trying to simplify √98, follow the step above to find that 98 ÷ 2 = 49, so 98 = 2 x 49. Rewrite "98" in the original square root using this information: √98 = √(2 x 49).

Simplify a Square Root Step 4
Simplify a Square Root Step 4

Step 4. Repeat with one of the remaining numbers

Before we can simplify the root, we continue to factorize until we've broken it 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 by yourself that is equal to 2 x 2." Obviously, that number is 2! With that goal in mind, let's repeat the steps above for our example problem, √(2 x 49):

  • The 2 is already maximally factored (in other words, it's one of those prime numbers from the list above). Let's ignore it for now and try to split the 49 instead.
  • 49 cannot be divided equally by 2, 3, or 5. You can test this with a calculator or by doing the division. Since these numbers don't yield whole results, let's just ignore them and keep trying.
  • 49 can be divided equally by 7. 49 ÷ 7 = 7, so 49 = 7 x 7.
  • Rewrite the problem: √(2 x 49) = √(2 x 7 x 7).
Simplify a Square Root Step 5
Simplify a Square Root Step 5

Step 5. Finish the simplification by "taking out" an integer

Once you break the problem down into two identical factors, you can turn it into an ordinary integer outside the square root. Leave all the other factors in it. For example, √(2 x 7 x 7) = √(2)√(7 x 7) = √(2) x 7 = 7√(2).

Even if it's possible to continue factoring, you don't have to once you've found two identical factors. For example, √(16) = √(4 x 4) = 4. If we continued factoring, we would end up with the same answer, but doing a bigger job. √(16) = √(4 x 4) = √(2 x 2 x 2 x 2) = √(2 x 2)√(2 x 2) = 2 x 2 = 4

Simplify a Square Root Step 6
Simplify a Square Root Step 6

Step 6. Multiply whole numbers if there is more than one

On some large square roots, you can simplify more than once. If that happens, multiply the integers to get to the final problem. Here is an example:

  • √180 = √(2 x 90).
  • √180 = √(2 x 2 x 45).
  • √180 = 2√45, but this can still be simplified.
  • √180 = 2√(3 x 15).
  • √180 = 2√(3 x 3 x 5).
  • √180 = (2)(3√5).
  • √180 = 6√5.
Simplify a Square Root Step 7
Simplify a Square Root Step 7

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

Some square roots are already in the simplest form. If you keep factoring until each term below the square root is a prime number (listed in one of the steps above) and no two numbers are the same, there's nothing you can do. You may have received 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. Also, they are all different, so it is not possible to "remove" an integer. √70 cannot be simplified.

Method 2 of 3: Knowing the Perfect Squares

Simplify a Square Root Step 8
Simplify a Square Root Step 8

Step 1. Memorize some perfect squares

Squared a number, or multiplied it by itself, creates a perfect square. For example, 25 is a perfect square because 5 x 5, or 52 is equal to 25. Memorizing at least the first ten perfect squares can help you quickly recognize and simplify perfect square roots. Here are the first 10 perfect squares:

  • 12 = 1.
  • 22 = 4.
  • 32 = 9.
  • 42 = 16.
  • 52 = 25.
  • 62 = 36.
  • 72 = 49.
  • 82 = 64.
  • 92 = 81.
  • 102 = 100.
Simplify a Square Root Step 9
Simplify a Square Root Step 9

Step 2. Find the square root of a perfect square

If you recognize a perfect square below a square root symbol, you can immediately make it its square root and get rid of the root symbol (√). For example, if you see the number 25 below the square root symbol, you already know that the answer is 5 because 25 is a perfect square. Here's the same list as above, this time going from square root to answer:

  • √1 = 1.
  • √4 = 2.
  • √9 = 3.
  • √16 = 4.
  • √25 = 5.
  • √36 = 6.
  • √49 = 7.
  • √64 = 8.
  • √81 = 9.
  • √100 = 10.
Simplify a Square Root Step 10
Simplify a Square Root Step 10

Step 3. Factor numbers into perfect squares

Use perfect squares to help you when following the factorization method of simplifying square roots. If you see a way to get a perfect square, it can save you time and effort. Here are some tips:

  • √50 = √(25 x 2) = 5√2. If the last two digits of a number end in 25, 50, or 75, you can always get 25.
  • √1700 = √(100 x 17) = 10√17. If the last two digits end in 00, you can always get 100.
  • √72 = √(9 x 8) = 3√8. Recognizing multiples of 9 is often helpful. Here's a trick for that: if, when adding all the digits of a number, the result is 9, then 9 is always a factor.
  • √12 = √(4 x 3) = 2√3. There's no special trick here, but it's generally easy to check if a small number is divisible by 4. Keep this in mind when looking for factors.
Simplify a Square Root Step 11
Simplify a Square Root Step 11

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

If the factors of a number contain more than one perfect square, move them all outside the radical symbol. If you find several perfect squares during the simplification process, move all of their square roots outside of the √ symbol and multiply them. For example, let's 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: Knowing the Terminology

Simplify a Square Root Step 12
Simplify a Square Root Step 12

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

For example, in problem √25, "√" is the symbol for the stem.

Simplify a Square Root Step 13
Simplify a Square Root Step 13

Step 2. Know that the root is the number inside the root symbol

You need to find the square root of this number. For example, in problem √25, "25" is the root.

Simplify a Square Root Step 14
Simplify a Square Root Step 14

Step 3. Know that the coefficient is the number outside the radical symbol

This is the number the square root is being multiplied by; it is to the left of the √ symbol. For example, in problem 7√2, "7" is the coefficient.

Simplify a Square Root Step 15
Simplify a Square Root Step 15

Step 4. Know that a factor is a number that divides another equally, leaving no remainder

For example, 2 is a factor of 8 because 8 ÷ 4 = 2, but 3 is not a factor of 8 because 8 ÷ 3 does not result in an integer. As another example: 5 is a factor of 25 because 5 x 5 = 25.

Simplify a Square Root Step 16
Simplify a Square Root Step 16

Step 5. Understand what is meant by simplifying a square root

This just means factoring out any perfect squares from the radicand, moving them to the left of the radical symbol and leaving the other factor inside the symbol. If the number is a perfect square, the radical symbol will disappear after you write the root. For example, √98 can be simplified to 7√2.

Tips

One way to find perfect square roots that factor into a number is to look through the list of perfect squares, starting with the next smallest number compared to its root. For example, when looking for the perfect square that fits 27, you can start at 25 and go down the list to 16, stopping at 9 when you find that is a factor of 27

Notices

  • Simplifying is not the same as evaluating. At no point in this process should you get a decimal point number!
  • Calculators can be useful for big numbers, but the more you practice doing it yourself, the easier it gets.

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