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The square root is a fundamental mathematical operation that is frequently encountered in various mathematical and scientific fields. It represents the inverse operation of squaring a number and finding the value that, when multiplied by itself, yields the original number. While square roots can sometimes appear complex or challenging to calculate, there are methods and techniques available to simplify them and make the process more manageable. In this article, we will explore different approaches to simplify square roots, providing step-by-step instructions and helpful tips to help you simplify square root expressions efficiently and accurately. By mastering these techniques, you will enhance your mathematical proficiency and gain the confidence to tackle more complex mathematical problems that involve square root calculations.
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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.”
Steps
Simplify the square root by factoring
- 2
- 3
- 5
- 7
- 11
- 13
- 17
- 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).
- 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.
- 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
- 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
- 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
- √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.
- 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
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.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 52 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 46,202 times.
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.”
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