Multiplication of Irrational Numbers(With Examples)

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Multiplication is one of the four fundamental operations in math. As you can perform multiplication with natural numbers, whole numbers, integers, and rational numbers, similarly you can perform this operation with irrational numbers.

Let’s understand how you can perform multiplication with irrational numbers.

Multiplication of Irrational Numbers

You can perform multiplication 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 multiply them:

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

You can multiply an irrational number with any real number, i.e., an irrational number can be multiplied by a natural number, whole number, integer, or a rational number. 

Note: 

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

Multiplication of Rational Number With Irrational Number

When a rational number is multiplied by an irrational number then their product 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 multiplication of irrational numbers with rational numbers.

Ex 1: Find the product of $3$ and $\sqrt {5}$.

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

In $3\sqrt{5}$, $3$ is a rational part and $\sqrt{5}$ is an irrational part.

You can also write $3\sqrt{5}$ as $\sqrt{3^{2}\times5}$ $\left(\text{Since, } 3 = \sqrt{3^{2}} = \sqrt{9} \right)$.

Therefore, $3\sqrt{5} = \sqrt{3^{2}\times5} = \sqrt{45}$

Note: $\sqrt{45}$ is also an irrational number.

Ex 2: Find the product of $2$ and $\sqrt[3] {7}$.

$2 \times \sqrt[3] {7} = 2\sqrt[3] {7}$.

$2\sqrt[3] {7}$ can also be written as $\sqrt[3] {2^{3} \times 7}$ $\left(\text{Since, } 2 = \sqrt[3]{2^{3}} = \sqrt[3]{8} \right)$.

Therefore, $2\sqrt[3] {7} = \sqrt[3] {2^{3} \times 7} = \sqrt[3] {8 \times 7} = \sqrt[3] {56}$.

Note: \sqrt[3] {56} is also an irrational number.

Multiplication of Irrational Number With Irrational Number

The product of an irrational number with another irrational number can be an irrational number or it can also be a rational number. 

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

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

Multiplication of Irrational Numbers With The Similar Root

When you multiply two or more irrational numbers with similar roots, then find the product of the numbers inside the roots.

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

Examples

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

Ex 1: Find the product of $\sqrt {2}$ and $\sqrt {3}$.

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

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

Note: A result is an irrational number.

Ex 2: Find the product of $\sqrt [5]{7}$ and $\sqrt [5]{2}$.

Here also the two irrational numbers have similar roots.

$\sqrt [5]{7} \times \sqrt [5]{2} = \sqrt [5]{7 \times 2} = \sqrt [5]{14}$.

Note: A result is an irrational number.

Ex 3: Find the product of $3\sqrt [3]{2}$ and $\sqrt [3]{5}$.

$3$ 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, $3\sqrt [3]{2} \times \sqrt [3]{5} = \sqrt [3]{54} \times \sqrt [3]{5} = \sqrt [3]{54 \times 5} = \sqrt [3]{270}$.

Note: A result is an irrational number.

Ex 4: Find the product of $\sqrt {2}$ and $\sqrt {8}$.

$\sqrt {2} \times \sqrt {8} = \sqrt {2 \times 8} = \sqrt {16} = \sqrt {4^{2}} = 4$.

Note: A result is a rational number.

Multiplication of Irrational Numbers With Different Roots

When you multiply two or more 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 multiply 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: Find the product of $\sqrt{3}$ and $\sqrt[3]{2}$.

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 multiply these numbers.

$\sqrt{3} \times \sqrt[3]{2} = \sqrt[6]{27} \times \sqrt[6]{4} = \sqrt[6]{27 \times 4} =  \sqrt[6]{108}$.

Note: A result is an irrational number.

Ex 2: Find the product of $\sqrt[3]{2}$ and $\sqrt[5]{3}$.

L.C.M. of radices $3$ and $5$ is $15$.

$\sqrt[3]{2} = \sqrt[15]{2^{5}} = \sqrt[15]{32}$

And,  $\sqrt[5]{3} = \sqrt[15]{3^{3}} = \sqrt[15]{27}$

Therefore, $\sqrt[3]{2} \times \sqrt[5]{3} = \sqrt[15]{32} \times \sqrt[15]{27} = \sqrt[15]{32 \times 27} = \sqrt[15]{864}$.

Note: A result is an irrational number.

Ex 3: Find the product of $\sqrt{2}$ and $\sqrt[4]{4}$.

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

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

Therefore, $\sqrt{2} \times \sqrt[4]{4} = \sqrt[4]{4} \times \sqrt[4]{4} = \sqrt[4]{4 \times 4} = \sqrt[4]{16} = \sqrt[4]{2^{4}} = 2$.

Note: A result is a rational number.

Conclusion

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

Practice Problems

Perform the following operations

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

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

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