All Square Roots Of 49

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Simplifying a foursquare root isn't as hard as information technology looks. To simplify a square root, you only accept to cistron the number and pull the roots of any perfect squares you observe out of the radical sign. In one case you've memorized a few common perfect squares and know how to factor a number, you'll be well on your mode to simplifying the square root.

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Understand factoring. The goal of simplifying a foursquare root is to rewrite it in a form that is easy to understand and to utilise in math problems. Factoring breaks down a large number into two or more smaller factors, for instance turning 9 into iii x 3. Once we find these factors, nosotros can rewrite the foursquare root in simpler form, sometimes even turning it into a normal integer. For example, √9 = √(3x3) = 3. Follow the steps below to learn this process for more complicated foursquare roots.^[1]
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Divide by the smallest prime number number possible. If the number under the square root is fifty-fifty, divide information technology past 2. If your number is odd, try dividing information technology by iii instead. If neither of these gives you a whole number, move down this list, testing the other primes until you get a whole number issue. You only need to examination the prime number numbers, since all other numbers have prime number numbers as their factors. For example, you lot don't need to examination four, because any number divisible by 4 is also divisible past 2, which yous already tried.^[2]
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- v
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- 17
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Rewrite the foursquare root as a multiplication problem. Keep everything underneath the square root sign, and don't forget to include both factors. For instance, if y'all're trying to simplify √98, follow the stride above to discover that 98 ÷ 2 = 49, so 98 = 2 ten 49. Rewrite the "98" in the original foursquare root using this data: √98 = √(2 x 49).^[3]
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Repeat with ane of the remaining numbers. Before we tin can simplify the square root, we keep factoring it until we've cleaved it down into ii identical parts. This makes sense if you call up about what a square root means: the term √(ii x 2) means "the number you tin can multiply with itself to equal 2 ten 2." Patently, this number is ii! With this goal in mind, let's echo the steps higher up for our example problem, √(ii 10 49):
- 2 is already factored as low as it will become. (In other words, it's one of those prime numbers on the list above.) We'll ignore this for now and endeavour to separate 49 instead.
- 49 can't be evenly divided past two, or past 3, or by 5. You can test this yourself using a calculator or long partitioning. Considering these don't give us nice, whole number results, we'll ignore them and go on trying.
- 49 can be evenly divided past seven. 49 ÷ 7 = 7, then 49 = 7 x vii.
- Rewrite the problem: √(2 x 49) = √(two x 7 x 7).
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Finish simplifying by "pulling out" an integer. Once you've broken the problem downwardly into two identical factors, you can plough that into a regular integer outside the square root. Go out all other factors inside the square root. For example, √(2 x 7 ten 7) = √(2)√(seven x 7) = √(2) x 7 = seven√(2).^[iv]
- Even if it'southward possible to keep factoring, y'all don't need to one time you've found two identical factors. For case, √(16) = √(4 x 4) = 4. If we kept on factoring, we'd end up with the same answer but have to do more work: √(xvi) = √(4 x 4) = √(ii ten 2 ten 2 x 2) = √(ii x 2)√(2 x 2) = 2 x 2 = 4.
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Multiply integers together if there are more than than one. With some big square roots, you can simplify more in one case. If this happens, multiply the integers together to get your final problem. Here'southward an case:
- √180 = √(2 x 90)
- √180 = √(2 x 2 x 45)
- √180 = ii√45, but this can still be simplified further.
- √180 = 2√(3 x xv)
- √180 = 2√(3 x 3 ten 5)
- √180 = (2)(three√5)
- √180 = 6√5
7
Write "cannot be simplified" if there are no two identical factors. Some square roots are already in simplest grade. If yous continue factoring until every term under the foursquare root is a prime (listed in one of the steps above), and no ii are the same, then there'due south null you tin do. You lot might take been given a play a joke on question! For example, permit's try to simplify √seventy:^[5]
- seventy = 35 x 2, so √70 = √(35 x ii)
- 35 = 7 x 5, so √(35 ten 2) = √(7 10 5 10 ii)
- All three of these numbers are prime, so they cannot be factored further. They're all different, then at that place'due south no way to "pull out" an integer. √seventy cannot be simplified.

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1
Memorize a few perfect squares. Squaring a number, or multiplying information technology past itself, creates a perfect square. For example, 25 is a perfect foursquare because five 10 5, or 5ⁱⁱ, equals 25. Memorizing at least the offset x perfect squares can assist you lot recognize and quickly simplify perfect square roots. Hither are the first x perfect squares:
- ane² = 1
- 2² = four
- three² = 9
- 4² = 16
- 5² = 25
- vi² = 36
- 7ⁱⁱ = 49
- eight² = 64
- 9^two = 81
- x^two = 100
2
Detect the square root of a perfect foursquare. If you lot recognize a perfect square under a square root symbol, you can immediately plow it into its square root and become rid of the radical sign (√). For example, if you see the number 25 under the square root sign, you know that the answer is 5 considering 25 is a perfect foursquare. Here's the same list as above, going from the square root to the reply:^[six]
- √1 = i
- √4 = 2
- √9 = 3
- √16 = 4
- √25 = 5
- √36 = 6
- √49 = vii
- √64 = eight
- √81 = 9
- √100 = 10
3
Factor numbers into perfect squares. Use the perfect squares to your reward when following the factor method of simplifying foursquare roots. If you lot observe a style to gene out a perfect foursquare, it can save you time and effort. Hither are some tips:^[7]
- √50 = √(25 ten two) = 5√ii. If the last two digits of a number stop in 25, l, or 75, yous can always cistron out 25.
- √1700 = √(100 x 17) = 10√17. If the last two digits end in 00, you tin always cistron out 100.
- √72 = √(9 x eight) = iii√8. Recognizing multiples of 9 is often helpful. There'southward a trick to it: if all digits in a number add up to nine, then nine is e'er a factor.
- √12 = √(4 x 3) = two√three. In that location'south no special flim-flam here, but it's normally piece of cake to check whether a small number is divisible by four. Keep this in mind when looking for factors.
iv
Factor a number with more than i perfect square. If the number'due south factors contain more than one perfect foursquare, move them all exterior the radical symbol. If you constitute multiple perfect squares during your simplification procedure, move all of their square roots to the outside of the √ symbol and multiply them together. For example, let's simplify √72:
- √72 = √(9 10 8)
- √72 = √(9 x 4 x 2)
- √72 = √(9) x √(four) x √(two)
- √72 = 3 10 2 x √2
- √72 = half dozen√2

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one

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

Know that the radicand is the number inside the radical symbol. You will need to find the square root of this number. For example, in the problem √25, "25" is the radicand.^[9]
iii

Know that the coefficient is the number exterior the radical symbol. This is the number that the square root is beingness multiplied by; this sits to the left of the √ symbol. For example, in the problem, seven√ii, "7" is the coefficient.
4

Know that a factor is a number that can exist evenly divided out of another number. For example, two is a factor of 8 because 8 ÷ 4 = 2, but 3 is not a cistron of 8 because 8÷three doesn't upshot in a whole number. As another example, v is a factor of 25 because 5 x 5 = 25.
five

Understand the significant of simplifying a square root. Simplifying a square root just 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 square, then the radical sign volition disappear once you write down its root. For case, √98 can be simplified to vii√two.

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Question

How do you simplify radical fractions?

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Staff Answer

Start by multiplying the numerator and denominator by a radical that will eliminate the radical in the denominator. For case, if your fraction is ii/√7, multiply past √seven/√7 to go ii√7/√49. This volition simplify to ii√7/7. If possible, simplify the fraction past dividing out whatever common factors in the numerator and denominator.
Question

What is the simplest radical form?

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Staff Respond

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

How practice you simplify square roots with variables?

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

wikiHow Staff Editor

Staff Answer

Employ the same procedure you would for simplifying square roots with numerical radicands. Bring whatever identical pairs of factors out of the radical to go the coefficient. For case, if y'all need to simplify √25a^iii, change it to √5×5×a×a×a. Factor out 5 and a^2 to get 5a^2√a.

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One mode to find perfect squares that factor into a number is to look through the list of perfect squares, beginning with the i that is the side by side smallest compared to your radicand, or the number under the square root sign. For example, when looking for the perfect square that goes into 27, you might get-go at 25 and go downward the listing to 16 and stop at 9, when you constitute the 1 that divides into 27.

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Calculators can be useful for large numbers, but the more you practice working this out on your own, the easier this volition get.

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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 information technology!

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Article 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 foursquare root of 98, the smallest prime number possible is 2. If you divide 98 by 2, yous get 49. Then, rewrite the foursquare root as a multiplication trouble under the square root sign. In this case, you'd rewrite the foursquare root every bit 2 × 49 under the foursquare root sign. From at that place, keep factoring the numbers until you take two identical factors. In this case, 49 divided by seven is 7. Rewrite the foursquare root every bit 2 × 7 × 7. Finally, once you have 2 identical numbers, movement them outside of the square root to make them a regular integer. So the simplified square root of 98 is 7 × the foursquare root of 2. However, if you factor the numbers within the square root equally much equally you lot tin can without getting two identical numbers, then your square root tin can't exist simplified! To learn other ways you can simplify a foursquare root, keep reading!

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