Square Root In Simplest Form

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Simplifying a square root isn't as hard equally it looks. To simplify a foursquare root, you but have to cistron the number and pull the roots of any perfect squares you find out of the radical sign. One time you've memorized a few common perfect squares and know how to factor a number, y'all'll be well on your way to simplifying the foursquare root.

1

Understand factoring. The goal of simplifying a square root is to rewrite it in a form that is easy to empathize and to use in math bug. Factoring breaks down a large number into two or more smaller factors, for case turning 9 into three x 3. Once we find these factors, nosotros can rewrite the square root in simpler class, sometimes even turning it into a normal integer. For example, √9 = √(3x3) = three. Follow the steps below to learn this process for more than complicated foursquare roots.^[ane]
2
Divide past the smallest prime number number possible. If the number under the square root is fifty-fifty, carve up it by 2. If your number is odd, try dividing it by three instead. If neither of these gives you a whole number, move down this list, testing the other primes until you get a whole number event. 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 iv, because whatsoever number divisible past iv is too divisible by 2, which yous already tried.^[2]
- two
- 3
- five
- 7
- xi
- thirteen
- 17
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3

Rewrite the square root every bit a multiplication problem. Proceed everything underneath the square root sign, and don't forget to include both factors. For example, if you lot're trying to simplify √98, follow the step above to find that 98 ÷ ii = 49, so 98 = 2 x 49. Rewrite the "98" in the original square root using this information: √98 = √(2 10 49).^[3]
4
Repeat with i of the remaining numbers. Before we can simplify the square root, nosotros go along factoring it until we've broken it down into two identical parts. This makes sense if yous call back about what a square root means: the term √(2 x 2) means "the number you can multiply with itself to equal 2 10 two." Obviously, this number is ii! With this goal in heed, let'southward repeat the steps above for our instance trouble, √(2 10 49):
- 2 is already factored as low as it will go. (In other words, it'due south i of those prime number numbers on the list above.) We'll ignore this for now and try to divide 49 instead.
- 49 can't be evenly divided past 2, or by 3, or past five. You tin can test this yourself using a calculator or long division. Considering these don't requite u.s. nice, whole number results, nosotros'll ignore them and go on trying.
- 49 can exist evenly divided past seven. 49 ÷ vii = vii, so 49 = 7 10 seven.
- Rewrite the problem: √(2 10 49) = √(ii x seven ten vii).
v
End simplifying by "pulling out" an integer. Once you've broken the problem downwards into two identical factors, you tin plough that into a regular integer exterior the square root. Get out all other factors inside the square root. For example, √(2 ten 7 x 7) = √(2)√(7 10 7) = √(2) 10 7 = seven√(two).^[4]
- Fifty-fifty if it'southward possible to continue factoring, yous don't need to once y'all've plant two identical factors. For example, √(16) = √(four 10 4) = 4. If we kept on factoring, we'd terminate upwards with the aforementioned respond but take to exercise more work: √(16) = √(4 ten 4) = √(ii x 2 x two x two) = √(2 x 2)√(2 x 2) = 2 x 2 = four.
6
Multiply integers together if there are more than one. With some large square roots, yous can simplify more than than once. If this happens, multiply the integers together to get your terminal problem. Here's an instance:
- √180 = √(2 x 90)
- √180 = √(ii x 2 x 45)
- √180 = 2√45, but this tin still be simplified further.
- √180 = 2√(iii x 15)
- √180 = 2√(3 x three x five)
- √180 = (2)(3√5)
- √180 = 6√5
7
Write "cannot be simplified" if at that place are no two identical factors. Some square roots are already in simplest grade. If you keep factoring until every term under the square root is a prime (listed in one of the steps above), and no 2 are the same, then there's nothing y'all can do. You might have been given a trick question! For example, let'due south try to simplify √70:^[5]
- 70 = 35 x 2, so √70 = √(35 x 2)
- 35 = 7 x 5, and then √(35 10 2) = √(7 x 5 ten two)
- All three of these numbers are prime number, so they cannot exist factored further. They're all different, and then in that location'due south no style to "pull out" an integer. √lxx cannot exist simplified.

one
Memorize a few perfect squares. Squaring a number, or multiplying it by itself, creates a perfect square. For example, 25 is a perfect foursquare because 5 x v, or 5², equals 25. Memorizing at least the showtime ten perfect squares tin can help you recognize and quickly simplify perfect square roots. Hither are the first x perfect squares:
- oneⁱⁱ = 1
- 2² = 4
- iii² = 9
- four² = 16
- v^two = 25
- vi² = 36
- 7² = 49
- viii² = 64
- 9² = 81
- 10² = 100
ii
Detect the square root of a perfect square. If you recognize a perfect foursquare under a square root symbol, yous can immediately turn it into its square root and become rid of the radical sign (√). For example, if you see the number 25 nether the square root sign, you know that the answer is 5 considering 25 is a perfect square. Here'southward the same list as above, going from the square root to the answer:^[6]
- √ane = 1
- √4 = 2
- √9 = 3
- √16 = 4
- √25 = 5
- √36 = 6
- √49 = 7
- √64 = 8
- √81 = ix
- √100 = 10
3
Gene numbers into perfect squares. Use the perfect squares to your advantage when following the gene method of simplifying square roots. If you notice a style to gene out a perfect square, information technology tin salve you time and effort. Here are some tips:^[7]
- √50 = √(25 ten 2) = v√2. If the final two digits of a number end in 25, fifty, or 75, you can always factor out 25.
- √1700 = √(100 ten 17) = 10√17. If the concluding two digits terminate in 00, you tin can always cistron out 100.
- √72 = √(9 x viii) = iii√8. Recognizing multiples of nine is ofttimes helpful. At that place's a pull a fast one on to it: if all digits in a number add upward to nine, and then nine is ever a cistron.
- √12 = √(4 x 3) = 2√three. There's no special pull a fast one on hither, simply it'due south usually easy to cheque whether a modest number is divisible by 4. Go along this in mind when looking for factors.
4
Factor a number with more than 1 perfect square. If the number'south factors contain more one perfect square, move them all outside the radical symbol. If you establish multiple perfect squares during your simplification process, movement all of their foursquare roots to the outside of the √ symbol and multiply them together. For instance, let's simplify √72:
- √72 = √(9 x 8)
- √72 = √(9 x 4 x 2)
- √72 = √(9) ten √(4) 10 √(2)
- √72 = 3 x 2 x √two
- √72 = half dozen√ii

1

Know that the radical symbol (√) is the foursquare root symbol. For instance, in the trouble, √25, "√" is the radical symbol.^[8]
2

Know that the radicand is the number within the radical symbol. You will need to detect the foursquare root of this number. For example, in the problem √25, "25" is the radicand.^[ix]
three

Know that the coefficient is the number outside the radical symbol. This is the number that the foursquare root is beingness multiplied past; this sits to the left of the √ symbol. For example, in the trouble, vii√two, "vii" is the coefficient.
4

Know that a gene is a number that can be evenly divided out of another number. For example, 2 is a cistron of eight because 8 ÷ 4 = 2, simply 3 is not a factor of 8 because 8÷three doesn't event in a whole number. As another instance, 5 is a factor of 25 because 5 x 5 = 25.
5

Understand the pregnant of simplifying a square root. Simplifying a foursquare root merely means factoring out any perfect squares from the radicand, moving them to the left of the radical symbol, and leaving the other gene inside the radical symbol. If the number is a perfect foursquare, then the radical sign will disappear once you write down its root. For instance, √98 can be simplified to 7√two.

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Question

How do yous simplify radical fractions?

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wikiHow Staff Editor

Staff Answer

Start by multiplying the numerator and denominator past a radical that will eliminate the radical in the denominator. For instance, if your fraction is 2/√7, multiply by √7/√seven to get two√7/√49. This will simplify to 2√vii/7. If possible, simplify the fraction by dividing out any common factors in the numerator and denominator.
Question

What is the simplest radical form?

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wikiHow Staff Editor

Staff Answer

If something is written in its simplest radical form, that means that you have already institute all possible roots and eliminated whatever radicals from the denominator of a fraction. A square root tin can't be simplified any further if there are no 2 identical factors remaining and every term under the radical symbol is a prime number.
Question

How practise you simplify square roots with variables?

This answer was written past i of our trained team of researchers who validated information technology for accuracy and comprehensiveness.

wikiHow Staff Editor

Staff Answer

Utilize the same procedure you would for simplifying square roots with numerical radicands. Bring any identical pairs of factors out of the radical to become the coefficient. For example, if you need to simplify √25a^3, change it to √5×five×a×a×a. Factor out 5 and a^ii to get 5a^2√a.

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One manner to discover perfect squares that factor into a number is to look through the list of perfect squares, beginning with the i that is the next smallest compared to your radicand, or the number under the square root sign. For example, when looking for the perfect foursquare that goes into 27, you might start at 25 and go downwards the listing to xvi and stop at 9, when you found the i that divides into 27.

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Calculators can be useful for big numbers, but the more yous practice working this out on your own, the easier this will go.

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Simplifying is not the same every bit evaluating. At no point in this process should you get a number with a decimal point in it!

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Commodity Summary X

To simplify a square root, start by dividing the square root by the smallest prime number possible. For case, if you're trying to find the square root of 98, the smallest prime number possible is 2. If you divide 98 by 2, you get 49. And then, rewrite the square root as a multiplication trouble under the foursquare root sign. In this case, you'd rewrite the square root as 2 × 49 nether the square root sign. From there, keep factoring the numbers until you have 2 identical factors. In this example, 49 divided past 7 is 7. Rewrite the square root as 2 × seven × vii. Finally, once you have two identical numbers, movement them outside of the square root to make them a regular integer. Then the simplified foursquare root of 98 is 7 × the square root of ii. However, if you factor the numbers inside the square root every bit much every bit you can without getting two identical numbers, so your square root can't be simplified! To acquire other ways you lot can simplify a square root, keep reading!

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