# 4 Ways to Divide Square Roots

Dividing square roots is basically the same as simplifying a fraction. Of course, the presence of square roots complicates the process a bit, but some rules allow us to work with fractions relatively simply. The key is to remember that it is necessary to divide coefficients by coefficients, and radicands by radicands. Also, you cannot have a square root in the denominator.

## Steps

### Method 1 of 4: Dividing Radicans

#### Step 1. Assemble the fraction

If the expression is not already assembled in fraction form, construct it that way. Doing this makes it easier to follow the steps necessary to carry out the division by the square root. Remember that the fraction bar is also the division bar.

• For example, if you are calculating 144÷36{displaystyle {sqrt {144}}\div {sqrt {36}}}

, reescreva o problema da seguinte forma: 14436{displaystyle {frac {sqrt {144}}{sqrt {36}}}}

#### Step 2. Use a radical sign

If the problem has a square root in the numerator and denominator, you can place both radicands over a single radical sign - a radicand is the number under the radical sign, or square root. Doing so will simplify the simplification process.

• For example, 14436{displaystyle {frac {sqrt {144}}{sqrt {36}}}}

pode ser reescrito por 14436{displaystyle {sqrt {frac {144}{36}}}}

#### Step 3. Divide the radicands

Divide the numbers just as you would any whole number. Remember to put the quotients under a new radical sign.

• For example, 14436=4{displaystyle {frac {144}{36}}=4}

, então 14436=4{displaystyle {sqrt {frac {144}{36}}}={sqrt {4}}}

#### Step 4. Simplify if necessary

If the root (or 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. For example, 25 is a perfect root because 5×5=25{displaystyle 5\times 5=25}

• Por exemplo, 4 é uma raiz perfeita, pois 2×2=4{displaystyle 2\times 2=4}
• . Portanto:

4{displaystyle {sqrt {4}}}

=2×2{displaystyle ={sqrt {2\times 2}}}

=2{displaystyle =2}

Sendo assim, 14436=4=2{displaystyle {frac {sqrt {144}}{sqrt {36}}}={sqrt {4}}=2}

### Método 2 de 4: Fatorando radicandos

#### Step 1. Express the problem as a fraction

The expression has probably already been written this way; otherwise, change it. Solving the problem as a fraction makes it easier to follow the necessary steps, especially when factoring square roots. Remember that the fraction bar is also the division bar.

• For example, if you are calculating 8÷36{displaystyle {sqrt {8}}\div {sqrt {36}}}

, reescreva o problema da seguinte forma: 836{displaystyle {frac {sqrt {8}}{sqrt {36}}}}

#### Step 2. Factor each root

Factor the number just as you would any whole number. Keep the factors under the radical sign.

• For example:

836=2×2×26×6{displaystyle {frac {sqrt {8}}{sqrt {36}}}={frac {sqrt {2\times 2\times 2}}{sqrt {6\times 6}}}}

#### Step 3. Simplify the numerator and denominator of the fraction

To simplify a square root, remove each factor that forms a perfect square. A perfect square is the result of a whole number multiplied by itself. Now the factor will become the coefficient outside the square root.

• For example:

2×2×26×6{displaystyle {frac {sqrt {{cancel {2\times 2\times }}2}}{sqrt {cancel {6\times 6}}}}}

226{displaystyle {frac {2{sqrt {2}}}{6}}}

Sendo assim, 836=226{displaystyle {frac {sqrt {8}}{sqrt {36}}}={frac {2{sqrt {2}}}{6}}}

#### Step 4. Rationalize the denominator if necessary

As a rule, an expression cannot have a square root in the denominator. If that happens, it needs to be rationalized. In other words, you need to cancel the square root in the denominator. To do this, multiply the numerator by the denominator of the fraction by the square root you need to cancel.

• For example, if the expression is 623{displaystyle {frac {6{sqrt {2}}}{sqrt {3}}}}

, é preciso multiplicar o numerador e denominador por 3{displaystyle {sqrt {3}}}

623×33{displaystyle {frac {6{sqrt {2}}}{sqrt {3}}}\times {frac {sqrt {3}}{sqrt {3}}}}

=62×33×3{displaystyle ={frac {6{sqrt {2}}\times {sqrt {3}}}{{sqrt {3}}\times {sqrt {3}}}}}

=669{displaystyle ={frac {6{sqrt {6}}}{sqrt {9}}}}

=663{displaystyle ={frac {6{sqrt {6}}}{3}}}

#### Step 5. Keep simplifying if necessary

Sometimes there will be a coefficient that cannot be simplified or reduced. Simplify integers in the numerator and denominator by simplifying any fraction.

• For example, 26{displaystyle {frac {2}{6}}}

pode ser reduzido para 13{displaystyle {frac {1}{3}}}

, então 226{displaystyle {frac {2{sqrt {2}}}{6}}}

pode ser reduzido para 123{displaystyle {frac {1{sqrt {2}}}{3}}}

, ou apenas 23{displaystyle {frac {sqrt {2}}{3}}}

### Método 3 de 4: Dividindo raízes quadradas com coeficientes

#### Step 1. Simplify the coefficients

Coefficients are numbers outside the radical sign. To simplify them, divide or reduce them, ignoring the square roots for now.

• For example, if you are calculating 432616{displaystyle {frac {4{sqrt {32}}}{6{sqrt {16}}}}}

, comece simplificando 46{displaystyle {frac {4}{6}}}

. Tanto o numerador quanto o denominador podem ser divididos por um fator de 2. Portanto, você pode reduzir: 46=23{displaystyle {frac {4}{6}}={frac {2}{3}}}

#### Step 2. Simplify the square roots

If the numerator is equally divisible by the denominator, just divide the radicands. Otherwise, simplify each square root normally.

• For example, since 32 is equally divisible by 16, you can divide the square roots:3216=2{displaystyle {sqrt {frac {32}{16}}}={sqrt {2}}}

#### Step 3. Multiply the simplified coefficient(s) by the simplified square root

Remember that it is not possible to have a square root in a denominator; then, when multiplying a fraction by a square root, put the square root in the numerator.

• For example, 23×2=223{displaystyle {frac {2}{3}}\times {sqrt {2}}={frac {2{sqrt {2}}}{3}}}

#### Step 4. Cancel the square root in the denominator, if necessary

The procedure is known as denominator rationalization. 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.

• For example, if the expression is 4327{displaystyle {frac {4{sqrt {3}}}{2{sqrt {7}}}}}

, é preciso multiplicar o numerador e denominador por 7{displaystyle {sqrt {7}}}

437×77{displaystyle {frac {4{sqrt {3}}}{sqrt {7}}}\times {frac {sqrt {7}}{sqrt {7}}}}

=43×77×7{displaystyle ={frac {4{sqrt {3}}\times {sqrt {7}}}{{sqrt {7}}\times {sqrt {7}}}}}

=42149{displaystyle ={frac {4{sqrt {21}}}{sqrt {49}}}}

=4217{displaystyle ={frac {4{sqrt {21}}}{7}}}

### Método 4 de 4: Dividindo por um binômio com uma raiz quadrada

#### Step 1. Check if there is a binomial in the denominator

The denominator will be the divisor of the problem. A binomial is a two-term polynomial. This method only applies to square root division involving a binomial.

• For example, if you are calculating 15+2{displaystyle {frac {1}{5+{sqrt {2}}}}}

, existe um binômio no denominador, já que 5+2{displaystyle 5+{sqrt {2}}}

é um binômio de dois termos.

#### Step 2. Find the conjugate of the binomial

Conjugate pairs are binomials that have the same terms but opposite operations. Using a conjugate pair allows you to cancel a square root in the denominator.

• For example, 5+2{displaystyle 5+{sqrt {2}}}

e 5−2{displaystyle 5-{sqrt {2}}}

são pares conjugados, já que possuem os mesmos termos, mas operações opostas.

#### Step 3. Multiply the numerator and denominator by the conjugate of the denominator

Doing so allows you to cancel the square root, as the product of a conjugate pair is the difference of the square of each term in the binomial. That is, (a−b)(a+b)=a2−b2{displaystyle (a-b)(a+b)=a^{2}-b^{2}}

• Por exemplo:

15+2{displaystyle {frac {1}{5+{sqrt {2}}}}}

=1(5−2)(5+2)(5−2){displaystyle ={frac {1(5-{sqrt {2}})}{(5+{sqrt {2}})(5-{sqrt {2}})}}}

=5−2(52−(2)2{displaystyle ={frac {5-{sqrt {2}}}{(5^{2}-({sqrt {2}})^{2}}}}

=5+225−2{displaystyle ={frac {5+{sqrt {2}}}{25-2}}}

=5+223{displaystyle ={frac {5+{sqrt {2}}}{23}}}

Portanto, 15+2=5+223{displaystyle {frac {1}{5+{sqrt {2}}}}={frac {5+{sqrt {2}}}{23}}}

## Notices

• Never leave a radical in the denominator of a fraction; instead, simplify it or rationalize it.
• Never place or remove a decimal or mixed number in front of a radical; instead, change the fraction or simplify the entire expression.
• Never put a decimal in a fraction. That would be a fraction within a fraction.
• If the denominator includes any kind of addition or subtraction, use a conjugate pair method to remove radicals from the denominator.