# Division of surds | How to divide surds

In this section, we will discuss how to divide surds. For more details of surds, please visit the page an introduction to surds.

#### How to Divide Surds

The division of surds is one of the fundamental operations on surds. To divide one surd by another surd, we need to go through the following steps.

Step I: At first, we will express the quotient as a fraction.

Step II: We need to rationalize the denominator of the fraction.

Step III: Now, we will find the suitable surd-rationalizing factor for the denominator of the fraction.

Step IV: Next, we will multiply both the numerator and the denominator of the fraction by the above surd-rationalizing factor.

Step V: Simplifying the fraction we will get the desired answer ♣

Example: We will apply the above method to divide $\sqrt{3}$ by $\sqrt{2}.$

In the first step, the quotient will be expressed as a fraction in the following way:

$\dfrac{\sqrt{3}}{\sqrt{2}}$

Now we need to rationalize the denominator $\sqrt{2}.$ To do that we have to multiply both the numerator and the denominator of the above fraction by $\sqrt{2}.$ Doing that we get

$\dfrac{\sqrt{3}}{\sqrt{2}}$ $=\dfrac{\sqrt{3} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$

$=\dfrac{\sqrt{3 \times 2}}{\sqrt{2 \times 2}}$  $[\because \sqrt{a} \times \sqrt{b}=\sqrt{a \times b}]$

$=\dfrac{6}{2}$  $[\because \sqrt{a \times a}=a]$

So the desired answer is $=\frac{6}{2}$ ♣

#### Formulas of Surds Division

(i) $\sqrt{a} \div \sqrt{b}=\sqrt{\frac{a}{b}}$

(ii) $\sqrt[n]{a} \div \sqrt[n]{b}=\sqrt[n]{\frac{a}{b}}$

(iii) $x\sqrt{a} \div y\sqrt{b}=\frac{x}{y}\sqrt{\frac{a}{b}}$

(iv) $x\sqrt[n]{a} \div y\sqrt[n]{b}=\frac{x}{y}\sqrt[n]{\frac{a}{b}}$

(iv) $\sqrt[m]{a} \div \sqrt[n]{a}=a^{\frac{1}{m}-\frac{1}{n}}$

(v) $x\sqrt[m]{a} \div y\sqrt[n]{a}=\frac{x}{y} \times a^{\frac{1}{m}-\frac{1}{n}}$

#### Solved Problems on Division of Surds

Problem 1: Divide $3$ by $\sqrt{5}$

Solution:

$3 \div \sqrt{5}$

$=\dfrac{3}{\sqrt{5}}$

$=\dfrac{3 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}$

$=\dfrac{3\sqrt{5}}{5}$

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