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Dividing square roots is essentially simplifying a fraction. Of course, the presence of square roots makes the process a little more complicated, but certain rules allow us to work with fractions in a relatively simple way. The key thing to remember is that you must divide coefficients by coefficients, and radicands by radicands. You can also never have a square root in a denominator.
Steps
Method 1
Method 1 of 4:Dividing Radicands

1Set up a fraction. If your expression is not already set up like a fraction, rewrite it this way. This makes it easier to follow all the necessary steps when dividing by a square root. Remember that a fraction bar is also a division bar.^{[1] X Research source }
 For example, if you are calculating , rewrite the problem like this: .

2Use one radical sign. If your problem has a square root in the numerator and denominator, you can place both radicands under one radical sign.^{[2] X Research source } (A radicand is a number under a radical, or square root, sign.) This will simplify the simplifying process.
 For example, can be rewritten as .
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3Divide the radicands. Divide the numbers as you would any whole number. Make sure to place their quotient under a new radical sign.
 For example, , so .

4Simplify, if necessary. If the radicand is a perfect square, or if one of its factors is a perfect square, you need to simplify the expression. A perfect square is the product of a whole number multiplied by itself.^{[3] X Research source } For example, 25 is a perfect square, since .
 For example, 4 is a perfect square, since . Thus:
So, .
Advertisement  For example, 4 is a perfect square, since . Thus:
Method 2
Method 2 of 4:Factoring Radicands

1Express the problem as a fraction. You will likely already see the expression written this way. If not, change it. Solving the problem as a fraction makes it easier to follow all the necessary steps, especially when factoring the square roots. Recall that a fraction bar is also a division bar.^{[4] X Research source }
 For example, if you are calculating , rewrite the problem like this: .

2Factor each radicand. Factor the number as you would any whole number. Keep the factors under the radical signs.^{[5] X Research source }
 For example:
 For example:

3Simplify the numerator and denominator of the fraction. To simplify a square root, pull out any factors that make a perfect square. A perfect square is the result of a whole number multiplied by itself.^{[6] X Research source } The factor will now become a coefficient outside of the square root.
 For example:
So,
 For example:

4Rationalize the denominator, if necessary. As a rule, an expression cannot have a square root in the denominator. If your fraction has a square root in the denominator, you need to rationalize it. This means to cancel out the square root in the denominator. To do this, multiply the numerator and denominator of the fraction by the square root you need to cancel.^{[7] X Research source }
 For example, if your expression is , you need to multiply the numerator and denominator by to cancel the square root in the denominator:
.
 For example, if your expression is , you need to multiply the numerator and denominator by to cancel the square root in the denominator:

5Simplify further, if necessary. Sometimes you will be left with coefficients that can be simplified, or reduced. Simplify the whole numbers in the numerator and denominator as you would simplify any fraction.
 For example, reduces to , so reduces to , or simply .
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Method 3
Method 3 of 4:Dividing Square Roots with Coefficients

1Simplify the coefficients. These are the numbers outside of the radical sign. To simplify them, divide or reduce, ignoring the square roots for now.
 For example, if you are calculating , you would first simplify . The numerator and denominator can both be divided by a factor of 2. So, you can reduce: .

2Simplify the square roots. If the numerator is evenly divisible by the denominator, simply divide the radicands. If not, simplify each square root as you would any square root.
 For example, since 32 is evenly divisible by 16, you can divide the square roots:.

3Multiply the simplified coefficient(s) by the simplified square root. Remember that you cannot have a square root in a denominator, so when multiplying a fraction by a square root, place the square root in the numerator.
 For example, .

4Cancel the square root in the denominator, if necessary. This is called rationalizing the denominator. As a rule, an expression cannot have a square root in the denominator. To rationalize the denominator, multiply the numerator and denominator by the square root you need to cancel.^{[8] X Research source }
 For example, if your expression is , you need to multiply the numerator and denominator by to cancel the square root in the denominator:
Advertisement  For example, if your expression is , you need to multiply the numerator and denominator by to cancel the square root in the denominator:
Method 4
Method 4 of 4:Dividing by a Binomial with a Square Root

1Determine that you have a binomial in the denominator. The denominator will be the number in the problem you are dividing by. A binomial is a twotermed polynomial.^{[9] X Research source } This method only applies to division of square roots involving a binomial.
 For example, if you are calculating , you have a binomial in the denominator, since is a twotermed polynomial.

2Find the conjugate of the binomial. Conjugate pairs are binomials that have the same terms, but opposite operations.^{[10] X Research source } Using a conjugate pair will allow you to cancel the square root in the denominator.
 For example, and are conjugate pairs, since they have the same terms but opposite operations.

3Multiply the numerator and denominator by the denominator’s conjugate. Doing this will allow you to cancel the square root, because the product of a conjugate pair is the difference of the square of each term in the binomial.^{[11] X Research source } That is, .
 For example:
Thus, .
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Community Q&A

QuestionHow can I divide root 5 by 5?DonaganTop Answerer√5 / 5 cannot be rationalized, simplified, or reduced. All you could do is to divide √5 (2.236) by 5.

QuestionHow can divide 1 by sqrt*2?Community AnswerYou need to rationalize the denominator, since a denominator cannot have a square root. To do this, multiply the numerator and the denominator by sqrt*2: 1 x sqrt*2 = sqrt*2 sqrt*2 x sqrt*2 = 2. So it simplifies to sqrt*2/2.

QuestionHow do I divide root3/root3?DonaganTop AnswererAnything divided by itself is 1.

QuestionWhat is 6 square root 35?DonaganTop AnswererFirst find the square root of 35, then multiply by 6.

QuestionHow can I divide under root 35?Community AnswerYou can square the number you want to divide by and then put this new number directly under the 35 inside the square root. Generally, for dividing by x: (√35) / x = √35 / √(x^2 ) = √(35/x^2 ). As an example, if the number you want to divide by is 5, then: (√35) / 5 = √35 / √25 = √(35/25) = √(7/5) = √1.4.

QuestionWhat is the value of 9 over the square root of 15?Community AnswerYou need to rationalize the denominator, since you can't have a square root in the denominator. To do this, multiply the numerator and denominator by sqrt*15: 9 x sqrt*15 = 9sqrt*15 sqrt*15 x sqrt*15 = 15. So this simplifies to 9sqrt*15/15.

QuestionWhat is sqrt*2 + 2 divided by 2?Community Answersqrt*2 + 2/2 is already simplified, since there is a binomial in the numerator, and there is no factor common to each term in the numerator and denominator. To find the exact value, you will need to use a calculator.

QuestionIf you have a cubed root and are dividing by a number that is to the power 3, will the cubed root and the exponent cancel each other out?DonaganTop AnswererNo.

QuestionWhat is 5 root 3 divided by 3 root 3?DonaganTop Answerer5√3 / 3√3: the two √3 's cancel each other. What's left is 5/3, or 1 2/3.

QuestionWhat is the simplified version of 7 divided by the square root of 5?DonaganTop AnswererThe rule is to remove radicals from denominators. In this case, do it by multiplying by the fraction (√5/√5) (which is the equivalent of multiplying by 1 and thus not changing the value of the original expression). So the solution looks like this: (7 / √5)(√5 / √5) = (7√5) / 5. That could be expressed as (7/5)(√5).
Video
Tips
 Unlike adding and subtracting radicals, in division, the radicands do not need to be simplified to remove perfect squares before you begin. In fact, it is often better not to do so.Thanks!
 Many calculators have fraction buttons. Try entering the numerator's coefficient, hitting the fraction button, then entering the denominator's coefficient. When you hit the = sign, the calculator should rewrite the coefficients into lowest terms.Thanks!
 When working with square roots, improper fractions are better to work with than mixed numbers.Thanks!
Warnings
 Never put a decimal into a fraction. That would be a fraction within a fraction.Thanks!
 Never leave a radical in the denominator of a fraction, but instead simplify or rationalize it.Thanks!
 Never put or leave a decimal or a mixed number in front of a radical, but instead change either to a fraction and simplify the entire expression.Thanks!
 If your denominator includes any kind of addition or subtraction, then use a conjugate pair method to remove radical from the denominator.Thanks!
References
 ↑ http://www.mathgoodies.com/lessons/fractions/
 ↑ http://www.mathwarehouse.com/algebra/radicals/dividingsquareroots.php
 ↑ http://www.mathwarehouse.com/arithmetic/numbers/whatisaperfectsquare.php
 ↑ http://www.mathgoodies.com/lessons/fractions/
 ↑ http://www.purplemath.com/modules/radicals4.htm
 ↑ http://www.mathwarehouse.com/arithmetic/numbers/whatisaperfectsquare.php
 ↑ http://www.purplemath.com/modules/radicals5.htm
 ↑ http://www.purplemath.com/modules/radicals5.htm
 ↑ https://www.mathsisfun.com/definitions/binomial.html
About This Article
To divide square roots using radicands, set up the expression as a fraction using one radical sign. If your problem has a square root in the numerator and denominator, you can place both radicands under one radical sign. Then, divide the radicands just as you would whole numbers, making sure to place the radicand quotient under a new radical sign. If the radicand is a perfect square, or if one of its factors is a perfect square, simplify the expression to finalize your answer! To learn how to divide square roots by using coefficients, read on!