Division of Irrational Numbers(With Examples)

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The division is the reverse process of multiplication and is one of the four fundamental operations in math. The division of irrational numbers is a bit different as compared to numbers of other categories such as natural numbers, whole numbers, integers, and rational numbers. In the case of irrational numbers, it involves one more step called the rationalization of the denominator.

Let’s understand the procedure of division with irrational numbers.

Division of Irrational Numbers

You can perform division between rational and irrational numbers and also between two irrational numbers. The following pointers are to be kept in mind when you deal with real numbers and divide them:

• When the division is done on a rational and irrational number, the result is an irrational number.
• When the division is done on an irrational and irrational number, the result can be an irrational or rational number.

You can divide an irrational number with any real number, i.e., an irrational number can be divided by a natural number, whole number, integer, or a rational number and vice-versa. 

Note: 

• All natural numbers, whole numbers, and integers are rational numbers
• The rational numbers and irrational numbers together form a set of real numbers

Division of Rational and Irrational Numbers

Let’s now consider the division of rational and irrational numbers. When a rational number is divided by an irrational number or vice versa then their quotient can be written in either of the two ways.

• $a\sqrt[n]{b}$ $\left(a \text{ is a rational number and } \sqrt[n]{b} \text { is an irrational part}   \right)$
• $\sqrt[n]{c}$ $\left(\sqrt[n]{c} \text{ is an irrational number} \right)$

Examples

Let’s consider some examples to understand the division of irrational numbers with rational numbers.

Ex 1: Divide $5$ by $\sqrt {2}$.

$5 \div \sqrt{2} = \frac {5}{\sqrt{2}}$

Note: 

• The denominator of the fraction is an irrational number, therefore you need to rationalize the denominator. 
• To rationalize the denominator, multiply both the numerator and the denominator with $\sqrt{2}$.
• $\sqrt{2}$ is a rationalizing factor (RF) of $\sqrt{2}$.

Multiplying the numerator and the denominator with $\sqrt{2}$.

$\frac {5}{\sqrt{2}} \times \frac {\sqrt{2}}{\sqrt{2}} = \frac {5\sqrt{2}}{2}$.

Therefore, $5 \div \sqrt{2} = \frac {5\sqrt{2}}{2}$.

Ex 2: Divide $5$ by $\sqrt[3] {4}$.

$5 \div \sqrt[3] {4} = \frac {5}{\sqrt[3] {4}}$

Rationalizing factor of $\sqrt[3] {4}$ is $\sqrt[3] {4^{2}} = \sqrt[3] {16}$.

Multiplying the numerator and the denominator with $\sqrt[3] {16}$

$\frac {5}{\sqrt[3] {4}} \times \frac {\sqrt[3] {16}}{\sqrt[3] {16}} = \frac {5 \times \sqrt[3] {16}}{\sqrt[3] {4} \times \sqrt[3] {16}} = \frac {5\sqrt[3]{16}}{\sqrt[3]{64}} = \frac {5\sqrt[3]{16}}{\sqrt[3]{4^{3}}} = \frac {5\sqrt[3]{16}}{4}$.

Therefore, $5 \div \sqrt[3] {4} = \frac {5}{4} \sqrt[3]{16}$

Note: Results in both examples are irrational numbers.

Dividing Irrational Numbers

Next, let’s consider dividing irrational numbers. The quotient of an irrational number with another irrational number can be an irrational number or it can also be a rational number. 

Division of two or more irrational numbers can be of either of the two types

• Dividing irrational numbers with similar roots $\left( \sqrt[n]{a} \div \sqrt[n]{b} \right)$
• Dividing irrational numbers with different roots $\left( \sqrt[m]{a} \div \sqrt[n]{b} \right)$

Division of Irrational Numbers With The Similar Root

When you divide two irrational numbers with similar roots, then find the quotient of the numbers inside the roots.

Mathematically, it can be expressed as $\sqrt [n]{a} \div \sqrt [n]{b} = \sqrt [n]{a \div b} = \sqrt[n]{\frac {a}{b}}$.

Examples

To understand the division of two or more irrational numbers, let’s consider some examples.

Ex 1: Divide $\sqrt {2}$ by $\sqrt {3}$.

Notice that the two irrational numbers have similar roots, so they can be divided directly.

$\sqrt {2} \div \sqrt {3} = \frac {\sqrt{2}}{\sqrt{3}}$.

The denominator of the fraction is an irrational number, therefore, it is rationalized by multiplying both the numerator and the denominator with $\sqrt{3}$.

$\frac {\sqrt{2}}{\sqrt{3}} = \frac {\sqrt{2}}{\sqrt{3}} \times \frac {\sqrt{3}}{\sqrt{3}} = \frac {\sqrt{2}\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}} = \frac {\sqrt{6}}{3}$.

Therefore, $\sqrt {2} \div \sqrt {3} = \frac {\sqrt{6}}{3}$

Ex 2: Divide of $\sqrt [5]{11}$ by $\sqrt [5]{3}$.

Here also the two irrational numbers have similar roots.

$\sqrt [5]{11} \div \sqrt [5]{3} = \frac {\sqrt [5]{11}}{\sqrt [5]{3}}$.

Now, rationalize the denominator by multiplying both the numerator and the denominator with $\sqrt[5]{3^{4}} = \sqrt[5]{81}$

$\frac {\sqrt [5]{11}}{\sqrt [5]{3}} = \frac {\sqrt [5]{11}}{\sqrt [5]{3}} \times \frac {\sqrt[5]{81}}{\sqrt[5]{81}} = \frac {\sqrt [5]{11} \times \sqrt[5]{81}}{\sqrt [5]{3} \times \sqrt[5]{81}} = \frac {\sqrt[5]{11 \times 81}}{\sqrt[5]{3 \times 81}} = \frac {\sqrt[5]{891}}{\sqrt[5]{243}} = \frac {\sqrt[5]{891}}{\sqrt[5]{3^{5}}} = \frac {\sqrt[5]{891}}{3}$

Ex 3: Divide $3\sqrt [3]{2}$ by $\sqrt [3]{5}$.

$3$ can be written as $\sqrt[3]{3^{3}} = \sqrt[3]{27}$

Therefore, $3\sqrt [3]{2}$ becomes $\sqrt [3]{27 \times 2} = \sqrt [3]{54}$.

And, $3\sqrt [3]{2} \div \sqrt [3]{5} = \sqrt [3]{54} \div \sqrt [3]{5} = \frac {\sqrt[3]{54}}{\sqrt[3]{5}}$.

Rationalizing factor of $\sqrt[3]{5}$ is $\sqrt[3]{5^{2}} = \sqrt[3]{25}$

Multiplying both the numerator and the denominator with $\sqrt[3]{25}$, we get $\frac {\sqrt[3]{54}}{\sqrt[3]{5}} \times \frac {\sqrt[3]{25}}{\sqrt[3]{25}} = \frac {\sqrt[3]{54} \times \sqrt[3]{25}}{\sqrt[3]{5} \times \sqrt[3]{25}} = \frac {\sqrt[3]{1350}}{\sqrt[3]{125}} = \frac {\sqrt[3]{1350}}{\sqrt[3]{5^{3}}} = \frac {\sqrt[3]{1350}}{5} = \frac {\sqrt[3]{27 \times 50}}{5} = \frac {\sqrt[3]{27} \times \sqrt[3]{50}}{5} = \frac {\sqrt[3]{3^{3}} \times \sqrt[3]{50}}{5} = \frac {3 \times \sqrt[3]{50}}{5} = \frac {3}{5}\sqrt[3]{50}$

Therefore, $3\sqrt [3]{2} \div \sqrt [3]{5} = \frac {3}{5}\sqrt[3]{50}$.

Ex 4: Divide $\sqrt {2}$ by $\sqrt {8}$.

$\sqrt {8} = \sqrt {4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

$\sqrt {2} \div \sqrt{8} = \sqrt {2} \div 2\sqrt{2} = \frac {\sqrt{2}}{2\sqrt{2}} = \frac {1}{2}$.

Division of Irrational Numbers With Different Roots

When you divide two irrational numbers with different roots, then the first step is to convert all the irrational numbers with similar roots and then use the procedure to divide the irrational numbers with similar roots.

To convert irrational numbers with different roots, take L.C.M. of the radices (singular radix) of the roots. The L.C.M. will be the radix of the root of the result.

Note: The radix of $\sqrt{}$ is $2$, i.e., $\sqrt{}$ and $\sqrt[2]{}$ are one and the same. $\sqrt{5} = \sqrt[2]{5}$.

Examples

To understand the process, let’s consider some examples.

Ex 1: Divide $\sqrt{3}$ by $\sqrt[3]{2}$.

Radix of $\sqrt{3}$ is $2$ and that of $\sqrt[3]{2}$ is $3$.

L.C.M. of $2$ and $3$ is $6$.

$\sqrt{3}  = \sqrt[6]{3^{3}} = \sqrt[6]{27}$ and $\sqrt[3]{2} = \sqrt[6]{2^{2}} = \sqrt[6]{4}$

Now, the radices of the two irrational numbers are the same, so we can divide these numbers.

$\sqrt{3} \div \sqrt[3]{2} = \sqrt[6]{27} \div \sqrt[6]{4} = \frac {\sqrt[6]{27}}{\sqrt[6]{4}} = \frac {\sqrt[6]{27}}{\sqrt[6]{2^{2}}}$.

Rationalizing factor of $\sqrt[6]{2^{2}}$ is $\sqrt[6]{2^{4}} = \sqrt[6]{16}$.

Multiplying the numerator and the denominator with $\sqrt[6]{16}$.

$\frac {\sqrt[6]{27}}{\sqrt[6]{2^{2}}} = \frac {\sqrt[6]{27}}{\sqrt[6]{2^{2}}} \times \frac {\sqrt[6]{16}}{\sqrt[6]{16}} = \frac {\sqrt[6]{27} \times \sqrt[6]{16}}{\sqrt[6]{2^{2}} \times \sqrt[6]{16}} = \frac {\sqrt[6]{27 \times 16}}{\sqrt[6]{4 \times 16}} = \frac {\sqrt[6]{432}}{\sqrt[6]{64}} = \frac {\sqrt[6]{432}}{\sqrt[6]{2^{6}}} = \frac {\sqrt[6]{432}}{2}$.

Therefore, $\sqrt{3} \div \sqrt[3]{2} = \frac {\sqrt[6]{432}}{2}$. 

Note: A result is an irrational number.

Ex 2: Divide $\sqrt{2}$ by $\sqrt[4]{4}$.

L.C.M. of the radices $2$ and $4$ is $4$.

$\sqrt{2} = \sqrt[4]{2^{2}} = \sqrt[4]{4}$

Therefore, $\sqrt{2} \div \sqrt[4]{4} = \sqrt[4]{4} \div \sqrt[4]{4} = \frac {\sqrt[4]{4}}{\sqrt[4]{4}} = 1$.

Note: A result is a rational number.

Practice Problems

Perform the following operations

• $4 \div \sqrt{3}$
• $2 \div \sqrt[4]{3}$
• $4 \div \sqrt[3]{5}$
• $\sqrt{5} \div \sqrt{3}$
• $\sqrt[5]{2} \div \sqrt[5]{3}$
• $\sqrt{7} \div \sqrt[3]{2}$
• $\sqrt[3]{3} \div \sqrt{9}$
• $\sqrt[4]{2} \div \sqrt[5]{3}$

FAQs

Conclusion

You can divide an irrational number with a rational number or with an irrational number. When you divide an irrational by a rational number, then the result is always an irrational number, whereas the quotient of two irrational numbers can be an irrational or rational number.

Perform the following operations

$4 \div \sqrt{3}$

$2 \div \sqrt[4]{3}$

$4 \div \sqrt[3]{5}$

$\sqrt{5} \div \sqrt{3}$

$\sqrt[5]{2} \div \sqrt[5]{3}$

$\sqrt{7} \div \sqrt[3]{2}$

$\sqrt[3]{3} \div \sqrt{9}$

$\sqrt[4]{2} \div \sqrt[5]{3}$

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)