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

Table of Contents

**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 Division of Surds**

(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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