How to Simplify Square Root

Simplifying the square root is not difficult, we just need to split the lower root into factors, of which at least one factor is the square, and then extract the square root value of the prime number from the sign. that side. Once you’ve memorized a few common squares and know how to factor them, reducing square roots will be “as easy as candy.”

Table of Contents

Simplify the square root by factoring

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Understand what factor analysis is. The goal of square root reduction is to rewrite it in a simpler and easier to understand form to solve math questions. Factorization is a way of dividing a larger number into many smaller factors , for example splitting 9 into 3 x 3. Once we have found the factors of the number in question, we can rewrite the square root of the number. that into a simpler form, maybe even an integer. For example √9 = √(3×3) = 3. The steps below will show you the process of reducing more complex square roots.

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Divide the number under the root by the smallest possible prime. If the base part is even, divide by two. If it’s an odd number, try to see if it’s divisible by 3. In case the number below the root is not divisible by both 2 and 3, proceed to divide by the next prime numbers in the list below until you find the smallest prime divisor of the number under the root. We only consider prime numbers because all other numbers can be factored as the product of a prime number. For example, we’re not going to divide the bottom by 4, because any number that’s divisible by 4 is also divisible by 2.

2
3
5
7
11
13
17

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Rewrite the square root as a multiplication problem. Keep all the factors under the root sign. For example, when we reduce √98, we find that 98 ÷ 2 = 49, so 98 = 2 x 49. So we can rewrite it as: √98 = √(2 x 49).

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Repeat the above steps with the remaining multipliers. Before reducing the square root in question, we need to multiply the factors until we get the result of the analysis as two identical numbers. Recalling the meaning of the square root, this makes perfect sense: since √(2 x 2) means “the number that, when multiplied by itself, is 2 x 2.” And obviously in this case it’s the number 2. Similarly, we repeat these steps with our example √(2 x 49):

We have factored 2. (In other words, this is one of the prime numbers listed above). Therefore, we will ignore this number and continue to split 49 into smaller factors.
49 is not divisible by 2, 3, or 5. We can check by using a calculator or doing division. Since the result of dividing 49 by 2, 3 or 5 does not give us an integer, we will ignore these numbers and continue to divide.
49 is divisible by 7. We have 49 ÷ 7 = 7, that is 49 = 7 x 7.
Rewrite the problem, we get: √(2 x 49) = √(2 x 7 x 7).

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“Draw” a number from the radical sign. Once we have factored the number in question, including two identical numbers, we can pull the number out of the radical sign. All other factors remain under the root sign. For example: √(2 x 7 x 7) = √(2)√(7 x 7) = √(2) x 7 = 7√(2).

We can stop the analysis when we have found two identical factors. For example √(16) = √(4 x 4) = 4. If we continue the analysis, the final result remains the same, except that we have to perform the division more times: √(16) = √(4 x 4) = √(2 x 2 x 2 x 2) = √(2 x 2)√(2 x 2) = 2 x 2 = 4.

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If the number of factors under the root is more than one, multiply them together. With large square roots, we can perform the reduction many times. In that case, multiplying the factors will give the final result. Consider the example below:

180 = (2 x 90)
180 = (2 x 2 x 45)
√180 = 2√45, but the remaining root can still be further factored into smaller factors
180 = 2√(3 x 15)
180 = 2√(3 x 3 x 5)
√180 = (2)(3√5)
√180 = 6√5

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Write “nonreducible” if the factorization does not give two numbers the same. Some square roots are already in their reduced form. If we continue the analysis until all the factors below the root are prime (mentioned in the previous steps) and no two numbers are the same, then we cannot simplify any further. Maybe the topic under consideration is just a tip! For example, let’s 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 numbers are prime, so we can’t simplify any further. In addition, these three numbers are all different, so we can’t pull any of the three numbers out of the radical sign. So √70 cannot be reduced any more.

Square number

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Memorize the perfect squares. Squaring a number, or in other words multiplying a number by itself, will give us a perfect square. For example, 25 is a perfect square because 5 x 5, i.e. 5 ² , is equal to 25. Try to memorize at least the first ten perfect squares because from that they can make it easier to identify the corresponding square root. . The first ten perfect squares are:

1 ² = 1
2 ² = 4
3 ² = 9
4 ² = 16
5 ² = 25
6 ² = 36
7 ² = 49
8 ² = 64
9 ² = 81
10 ² = 100
Find the square root of a perfect square. If we see a perfect square under the radical sign, we can convert that number to the form of the product of two identical numbers, thereby eliminating the radical sign. For example, when we see that the lower part of the root is 25, we know that the value of this square root is 5 because 25 is a perfect square and is equal to 5 x 5. Similarly, we have the value of the square root of all perfect squares. above as follows:

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

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Factoring into perfect squares. When reducing square roots, use perfect squares in the factoring step. If it is possible to separate a perfect square, the reduction will take less time. Here are some tips:

√50 = √(25 x 2) = 5√2. If the last two digits of the number in question are 25, 50 or 75, we can always separate 25 from that number.
√1700 = √(100 x 17) = 10√17. If the last two digits of the number in question are 00, we can always separate 100 from that number.
√72 = √(9 x 8) = 3√8. Knowing multiples of 9 also helps a lot when factoring. The trick to recognize multiples of 9 is as follows: if the sum of all the digits of the number in question is equal to 9 or divisible by 9, then the number is divisible by 9.
√12 = √(4 x 3) = 2√3. There is no trick to tell if a number is divisible by 4, but for very large numbers, the division by 4 is not too complicated. Keep this square number in mind when factoring.

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Analyze some product of many perfect squares. If the number in question is the product of more than one perfect square, we can put them all outside the radical sign. During square root reduction, if the result of the factorization has many perfect squares, we remove their square roots from the sign of the root and multiply them together. Shortened example √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

Terms

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The sign (√) is the square root sign. For example, in the problem √25, “√” is the radical sign.

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The number under the root is the number written under the radical sign. We need to find the square root of that number. For example, for √25, “25” is the number under the root.

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The root factor is the number outside the radical sign. This is the number multiplied by the square root and is to the left of the root sign. For example with 7√2, then “7” is the root factor.

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The result of a divisibility is called the multiplier. For example, 2 is a factor of 8 because 8 ÷ 4 = 2, 3 is not a factor of 8 because 8÷3 does not result in an integer. For example, 5 is a factor of 25 because 5 x 5 = 25.

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Meaning of square root reduction. Square root reduction is the separation of the perfect squares from the number below the root, removing the square root of those squares from the root sign, and keeping the remaining factor under the root sign. If the number under the root is a perfect square, then after reduction we will eliminate the radical sign. For example, √98 can be reduced to 7√2.

Advice

One way to factor a perfect square is to go through a list of perfect squares, starting with the number closest to the root and stopping when you find a number that is a divisor of the number below the root. . For example, when looking for a perfect square that can be split from 27, you would start at 25 then go to 16 and stop at 9 because this is a divisor of 27.
We need to find the number that, when multiplied by itself, results in a number under the radical sign. For example the square root of 25 is 5 because if we take 5 x 5 we get 25. Easy as candy!

Warning

Calculators are quite useful in case you need to deal with large numbers, but the more you try to practice this form yourself, the easier it will be to reduce the square root.
Reduction and estimation are not the same. The square root reduction cannot result in a decimal number.

Steps

Simplify the square root by factoring

Square number

Terms

Advice

Warning

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