The nth root of a number – Meaning & Properties

 

When we talk about the root of a number, the square roots and the cube roots strike our minds. But, it is to be noted that the roots are not limited to the square roots and the cube roots. For eg- 2 is the fourth root of 16.

In the case of the square root and cube root which is denoted by √ and ∛ respectively, 2 and 3 are their degrees. So, ‘n’ is the degree of the root and it must be a positive number. It is denoted by \sqrt[n]{x}.

Finding the nth root of a number

When the nth root is multiplied n times by itself, we get the original value of a number. It only means that if \sqrt[n]{x}= y, then x=y^{n} where y is the nth root of x.

For eg – \sqrt[5]{32}, we have to find that value which when multiplied 5 times by itself yields the result 32.

By using the prime factorisation, we get 2^{5} so 2 is the 5th root of 32 i.e, the nth root, here n=5.

When the nth root of a number is raised to the power ‘n’ i.e., \sqrt[n]{x^{n}}, we get the following values-

When n is an even number and x ≥ 0, then \sqrt[n]{x^{n}}=x. For eg – \sqrt[2]{2^{2}}=2.

When n is an odd number and x ≥ 0, then \sqrt[n]{x^{n}}=x. For eg – \sqrt[3]{2^{3}}=2.

When n is an odd number and x < 0, then \sqrt[n]{x^{n}}=x. For eg – \sqrt[3]{-2^{3}}=-2.

When n is an even number and x < 0, then \sqrt[n]{x^{n}}=|x|. For eg – \sqrt[2]{-2^{2}}=|2|=2.

We have studied that  (-)\times (-) = (+). This is why we add the modulus symbol(I I) with the value so that all negative values will become positive.

 

Properties of the nth root of a number

Property 1

We can separate the multiplications under the nth root  – \sqrt[n]{p q}=\sqrt[n]{p} \times \sqrt[n]{q}.

This property can be explained with the help of an example – 

\sqrt[3]{64} =\sqrt[3]{8} \times \sqrt[3]{8}
=\sqrt[3]{2}^{3} \times \sqrt[3]{2^{3}}
=2 \times 2
=4

 

Property 2

Like multiplications, we can separate the divisions under the nth root – \sqrt[n]{\frac{p}{q}}=\frac{\sqrt[n]{p}}{\sqrt[n]{q}}        where p ≥ 0 and q > 0.

Let us try to prove it with the help of an example-

\sqrt[3]{\frac{512}{64}} =\frac{\sqrt[3]{512}}{\sqrt[3]{64}}

=\frac{\sqrt[3]{8^{3}}}{\sqrt[3]{4^{3}}} =\frac{8}{4}

 =2

So, it is proved that divisions can be separated under the root.

Property 3

The exponential form of the nth root of a number can be written as x^{\frac{1}{n}} . For eg – \sqrt[7]{2187}=(2187)^{\frac{1}{7}} which is equal to 3.

Property 4

The nth root of a number to the power mth \sqrt[n]{x^{m}}can be written as(\sqrt[n]{x})^{\mathrm{m}}.

For eg  \sqrt[2]{4^{4}}=\sqrt[2]{256}=16. Where \sqrt[2]{4^{4}} can be written as (\sqrt[2]{4})^{4}=2^{4}=16.

Note: Is \sqrt[n]{p \pm q}=\sqrt[n]{p} \pm \sqrt[n]{q} ?

The answer is no.

Let us try to prove it with the help of an example-

\sqrt[2]{100} can be written as \sqrt[2]{64+36}

We know that 10 is the square root of 100.

But \sqrt[2]{64}+\sqrt[2]{36}=8+6=14

14 ≠ 10.

Hence, \sqrt[n]{p \pm q} \neq \sqrt[n]{p} \pm \sqrt[n]{q}.