Content

• Steps
• Method 1 of 4: Dividing radical expressions
• Method 2 of 4: Factoring the Radical Expression
• Method 3 of 4: Multiplying Square Roots
• Method 4 of 4: Division by a square root binomial
• Tips
• Warnings

Dividing square roots simplifies the fraction. Having square roots complicates the solution a little, but some rules make it relatively easy to work with fractions. The main thing to remember is that factors are divided by factors, and radical expressions by radical expressions. Also, the square root can be in the denominator.

Steps

Method 1 of 4: Dividing radical expressions

1. 1 Write down the fraction. If the expression is not a fraction, rewrite it that way. This makes it easier to follow the process of dividing square roots. Remember that the horizontal bar represents the division sign.
   • For example, given the expression ${ displaystyle { sqrt {144}} div { sqrt {36}}}$, rewrite it like this: ${ displaystyle { frac { sqrt {144}} { sqrt {36}}}}$.
2. 2 Use one root sign. If both the numerator and the denominator of the fraction have square roots, write their radical expressions under one root sign to simplify the solution process. A radical expression is an expression (or just a number) that is under the root sign.
   • For example, the fraction ${ displaystyle { frac { sqrt {144}} { sqrt {36}}}}$ can be written like this: ${ displaystyle { sqrt { frac {144} {36}}}}$.
3. 3 Divide the radical expression. Divide one number by another (as usual), and write the result under the root sign.
   • For example, ${ displaystyle { frac {144} {36}} = 4}$, so: ${ displaystyle { sqrt { frac {144} {36}}} = { sqrt {4}}}$.
4. 4 Simplify radical expression (if necessary). If the radical expression or one of its factors is a perfect square, simplify that expression. A complete square is a number that is the square of some integer. For example, 25 is a perfect square because ${ displaystyle 5 times 5 = 25}$.
   • For example, 4 is a perfect square because ${ displaystyle 2 times 2 = 4}$... Thus:
     ${ displaystyle { sqrt {4}}}$
     ${ displaystyle = { sqrt {2 times 2}}}$
     ${ displaystyle = 2}$
     So: ${ displaystyle { frac { sqrt {144}} { sqrt {36}}} = { sqrt {4}} = 2}$.

Method 2 of 4: Factoring the Radical Expression

1. 1 Write down the fraction. If the expression is not a fraction, rewrite it that way. This makes it easier to follow the process of dividing square roots, especially when factoring a radical expression. Remember that the horizontal bar represents the division sign.
   • For example, given the expression ${ displaystyle { sqrt {8}} div { sqrt {36}}}$, rewrite it like this: ${ displaystyle { frac { sqrt {8}} { sqrt {36}}}}$.
2. 2 Spread out into factors of each radical expression. The number under the root sign is factorized like any integer. Write down the factors under the root sign.
   • For example:
     ${ displaystyle { frac { sqrt {8}} { sqrt {36}}} = { frac { sqrt {2 times 2 times 2}} { sqrt {6 times 6}}}}$
3. 3 Simplify the numerator and denominator of the fraction. To do this, take out the factors, which are complete squares, from under the root sign. A complete square is a number that is the square of some integer. The factor of the radical expression will turn into a factor before the sign of the root.
   • For example:
     ${ displaystyle { frac { sqrt {{ cancel {2 times 2 times}} 2}} { sqrt { cancel {6 times 6}}}}}$
     ${ displaystyle { frac {2 { sqrt {2}}} {6}}}$
     Thus, ${ displaystyle { frac { sqrt {8}} { sqrt {36}}} = { frac {2 { sqrt {2}}} {6}}}$
4. 4 Get rid of the root in the denominator (rationalize the denominator). In mathematics, it is not customary to leave the root in the denominator. If the fraction has a square root in the denominator, get rid of it. To do this, multiply both the numerator and the denominator by the square root you want to get rid of.
   • For example, given the fraction ${ displaystyle { frac {6 { sqrt {2}}} { sqrt {3}}}}$, multiply the numerator and denominator by ${ displaystyle { sqrt {3}}}$to get rid of the root in the denominator:
     ${ displaystyle { frac {6 { sqrt {2}}} { sqrt {3}}} times { frac { sqrt {3}} { sqrt {3}}}}$
     ${ displaystyle = { frac {6 { sqrt {2}} times { sqrt {3}}} {{ sqrt {3}} times { sqrt {3}}}}}$
     ${ displaystyle = { frac {6 { sqrt {6}}} { sqrt {9}}}}$
     ${ displaystyle = { frac {6 { sqrt {6}}} {3}}}$.
5. 5 Simplify the resulting expression (if necessary). Sometimes the numerator and denominator of a fraction contain numbers that can be simplified (reduced). Simplify the whole numbers in the numerator and denominator as you simplify any fraction.
   • For example, ${ displaystyle { frac {2} {6}}}$ simplifies to ${ displaystyle { frac {1} {3}}}$; thus ${ displaystyle { frac {2 { sqrt {2}}} {6}}}$ simplifies to ${ displaystyle { frac {1 { sqrt {2}}} {3}}}$ = ${ displaystyle { frac { sqrt {2}} {3}}}$.

Method 3 of 4: Multiplying Square Roots

1. 1 Simplify the factors. The factor is the number that precedes the root sign. To simplify factors, divide or reduce them (do not touch radical expressions).
   • For example, given the expression ${ displaystyle { frac {4 { sqrt {32}}} {6 { sqrt {16}}}}}$, first simplify ${ displaystyle { frac {4} {6}}}$... The numerator and denominator can be divided by 2. Thus, the factors can be canceled:${ displaystyle { frac {4} {6}} = { frac {2} {3}}}$.
2. 2 Simplify square roots. If the numerator is evenly divisible by the denominator, do so; otherwise, simplify the radical expression like any other expression.
   • For example, 32 is evenly divisible by 16, so:${ displaystyle { sqrt { frac {32} {16}}} = { sqrt {2}}}$
3. 3 Multiply simplified factors by simplified roots. Remember that it is best not to leave the root in the denominator, so multiply both the numerator and the denominator of the fraction by this root.
   • For example, ${ displaystyle { frac {2} {3}} times { sqrt {2}} = { frac {2 { sqrt {2}}} {3}}}$.
4. 4 Get rid of the root in the denominator if necessary (rationalize the denominator). In mathematics, it is not customary to leave the root in the denominator.Therefore, multiply both the numerator and denominator by the square root you want to get rid of.
   • For example, given the fraction ${ displaystyle { frac {4 { sqrt {3}}} {2 { sqrt {7}}}}}$, multiply the numerator and denominator by ${ displaystyle { sqrt {7}}}$to get rid of the root in the denominator:
     ${ displaystyle { frac {4 { sqrt {3}}} { sqrt {7}}} times { frac { sqrt {7}} { sqrt {7}}}}$
     ${ displaystyle = { frac {4 { sqrt {3}} times { sqrt {7}}} {{ sqrt {7}} times { sqrt {7}}}}}$
     ${ displaystyle = { frac {4 { sqrt {21}}} { sqrt {49}}}}$
     ${ displaystyle = { frac {4 { sqrt {21}}} {7}}}$

Method 4 of 4: Division by a square root binomial

1. 1 Determine that the denominator contains a binomial (binomial). The denominator is the divisor (expression or number below the line). A binomial (binomial) is an expression that includes two monomials. This method is applicable only when the problem contains a square root binomial.
   • For example, given the fraction ${ displaystyle { frac {1} {5 + { sqrt {2}}}}}$, the denominator contains a binomial, because the expression ${ displaystyle 5 + { sqrt {2}}}$ includes two monomials.
2. 2 Find the expression conjugated to the binomial. A conjugate binomial is a binomial with the same monomials, but with the opposite sign between them. Multiplying conjugate binomials will get rid of the root in the denominator.
   • For example, ${ displaystyle 5 + { sqrt {2}}}$ and ${ displaystyle 5 - { sqrt {2}}}$ are conjugate binomials because they include the same monomials, but with opposite signs between them.
3. 3 Multiply the numerator and denominator by the binomial conjugate to the binomial in the denominator. This will get rid of the square root, because the product of the conjugate binomials is equal to the difference of the squares of each binomial term. I.e ${ displaystyle (a-b) (a + b) = a ^ {2} -b ^ {2}}$.
   • For example:
     ${ displaystyle { frac {1} {5 + { sqrt {2}}}}}$
     ${ displaystyle = { frac {1 (5 - { sqrt {2}})} {(5 + { sqrt {2}}) (5 - { sqrt {2}})}}}$
     ${ displaystyle = { frac {5 - { sqrt {2}}} {(5 ^ {2} - ({ sqrt {2}}) ^ {2}}}}$
     ${ displaystyle = { frac {5 + { sqrt {2}}} {25-2}}}$
     ${ displaystyle = { frac {5 + { sqrt {2}}} {23}}}$
     Thus, ${ displaystyle { frac {1} {5 + { sqrt {2}}}} = { frac {5 + { sqrt {2}}} {23}}}$.

Tips

• Many calculators know how to work with fractions. Enter the number in the numerator, press the fraction key, and then enter the number in the denominator. Press "=" and the calculator will automatically simplify (reduce) the fraction.
• When working with square roots, it is better to convert a mixed number to an improper fraction.
• Unlike addition and subtraction of roots, when dividing them, radical expressions can not be simplified (due to complete squares); in fact, it is often best not to do it at all.

Warnings

• Never leave the root in the denominator of a fraction - simplify or rationalize it.
• The decimal fraction and mixed number are not placed in front of the root. Convert them to fractions and then simplify the resulting expression.
• Do not write the decimal in the denominator or numerator of a fraction; otherwise, you get a fraction in a fraction.
• If the denominator contains the sum or difference of two monomials, multiply this bin by its conjugate binomial to get rid of the root in the denominator.