Being able to calculate the square root of a number that is not a perfect square is not as difficult as it may seem. You have to factor the rooting and remove from the root any factor that is a perfect square. Once you have memorized the most common perfect squares, you will be able to easily simplify the square roots.
Steps
Part 1 of 3: Simplifying the Square Root with Factorization
Step 1. Learn about factoring
The goal, during the root simplification process, is to rewrite the problem in an easier form. The decomposition breaks down the number into smaller factors, for example the number 9 can be seen as the result of 3x3. Once the factors are identified, you can rewrite the square root into a simpler form and sometimes turn it into an integer. For example: √9 = √ (3x3) = 3. Follow the instructions to learn the procedure.
Step 2. Divide the number into the smallest possible prime factors
If the number under the root is even, divide it by 2. If the number is odd, try dividing it by 3. If you don't get an integer, continue with other prime numbers until division yields an integer quotient. You must use only the prime numbers as a divisor, since all the others are in turn the result of multiplying prime factors. For example you don't have to try to decompose a number by 4, as 4 is divisible by 2 (which you have already tested).
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- 5
- 7
- 11
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- 17
Step 3. Rewrite the square root as a multiplication
Keep all multiplication under the root sign without forgetting any factors. For example, if you need to simplify √98, follow the steps above and you will find that 98 ÷ 2 = 49, so 98 = 2 x 49. Rewrite "98" under the root sign, but as a multiplication: √98 = √ (2 x 49).
Step 4. Repeat the process with one of the two numbers
Before you can simplify the square root, you need to continue decomposing until you find two identical factors. This concept is easy to understand, if you think about what the square root means: the symbol √ (2 x 2) allows you to calculate "the number that multiplied by itself gives 2 x 2". Obviously, in this case it is 2! With that goal in mind, repeat the previous steps with the problem: √ (2 x 49):
- 2 is a prime number that cannot be further broken down. Ignore it and take care of 49.
- 49 is not divisible by 2, 3 or 5. You can check this with the calculator or a division by column. Since these factors do not give an integer quotient, ignore them and proceed further.
- 49 can be divided by 7. 49 ÷ 7 = 7, so 49 = 7 x 7.
- Rewrite the problem: √ (2 x 49) = √ (2 x 7 x 7).
Step 5. Finish the simplification by "extracting" an integer
Once you have broken down the problem into identical factors, you can extract an integer from the root symbol while leaving the other factors inside. For example: √ (2 x 7 x 7) = √ (2) √ (7 x 7) = √ (2) x 7 = 7√ (2).
While it is possible to continue breaking it down, it is not necessary to do so when you have found two identical numbers. For example: √ (16) = √ (4 x 4) = 4. If you continue with the decomposition you will get the same solution but with more work: √ (16) = √ (4 x 4) = √ (2 x 2 x 2 x 2) = √ (2 x 2) √ (2 x 2) = 2 x 2 = 4
Step 6. If there are more than one, multiply the integers together
When dealing with large square roots, you can simplify them into multiple factors. When this happens, you have to multiply the integers you extract from the root sign. Here is an example:
- √180 = √ (2 x 90)
- √180 = √ (2 x 2 x 45)
- √180 = 2√45, which 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. If you do not find identical factors, end the problem with the words "no further simplification possible"
Some square roots are already in minimal form. If, after reducing the number into prime factors, you do not find two equal numbers, then there is nothing you can do. The root that has been assigned to you cannot be simplified. 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 are prime and cannot be broken down. They are all different from each other and you cannot "extract" any integers. √70 cannot be simplified.
Part 2 of 3: Knowing the Perfect Squares
Step 1. Memorize some perfect squares and their square roots
Squaring a number (i.e. multiplying it by itself) results in a perfect square (for example, 25 is a perfect square because 5x5, or 52, makes 25). It is a good thing to be familiar with at least the first 10 perfect squares and their square roots, as this will allow you to simplify more complicated square roots with less difficulty. Here are the top 10:
- 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 a perfect square
The only thing you need to do is remove the root sign (√) and write the corresponding value. If you've memorized the first 10 perfect squares it won't be a problem. For example, if under the root sign there is the number 25, you know that the solution is 5 since 25 is its perfect square:
- √1 = 1
- √4 = 2
- √9 = 3
- √16 = 4
- √25 = 5
- √36 = 6
- √49 = 7
- √64 = 8
- √81 = 9
- √100 = 10
Step 3. Divide the numbers into factors which are themselves perfect squares
Take advantage of perfect squares when using the factorization method to simplify the roots. If you notice that one of the factors is also a perfect square, you will save a lot of time and effort. Here are some useful tips:
- √50 = √ (25 x 2) = 5√2. If the last two digits of a number are 25, 50 or 75 you can always extract the factor 25.
- √1700 = √ (100 x 17) = 10√17. If the last two digits are 00, you can always extract the factor 100.
- √72 = √ (9 x 8) = 3√8. Recognizing multiples of 9 is not easy. Here's a trick: if the sum of all digits in the number equals nine, then 9 is a factor.
- √12 = √ (4 x 3) = 2√3. There are no tricks for this case, but it's not hard to tell if a small number is divisible by 4. Remember this when looking for factors.
Step 4. Factor a number with more than one perfect square
If the number contains many factors that are at the same time perfect squares, you have to extract them from the root. In this case you have to remove them from the radical (√) and multiply them. Here is the example of √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
Part 3 of 3: Know the Terminology
Step 1. The radical (√) is the square root symbol
For example, in the problem √25, "√" is the radical.
Step 2. The radicand is the number under the root symbol
It is the value whose square root you need to find. For example in √25, "25" is the rooting.
Step 3. The coefficient is the number outside the root symbol
Indicates the number of times the root must be multiplied and is to the left of it. In 7√2, "7" is the coefficient.
Step 4. Factors are the numbers that divide the rooting into integer values
For example 2 is a factor of 8 because 8 ÷ 2 = 4, but 3 is not a factor of 8 because 8 ÷ 3 does not give an integer as a quotient. Instead 5 is a factor of 25 because 5 x 5 = 25.
Step 5. Understand the meaning of simplification
This is an operation that allows you to remove from the root sign every factor of the rooting which is a perfect square, leaving inside all the factors that are not. If the radicand is a perfect square, the root sign disappears and you have to write the root value. For example √98 can be simplified to 7√2.
Advice
One way to find a perfect square of your rooting is to check the list of perfect squares, starting with the smaller one than your rooting. For example, if you are looking for the perfect square of 27 you should start at 25 and then go down to 16 and stop at 9, when you find what 27 is divisible by
Warnings
- Simplifying is not the same as dividing. You shouldn't end up with a decimal point at any stage of the process!
- The calculator is useful when you have to work with large numbers, however the more you practice the calculations in mind, the easier the process will become.
