nth Root of a

We will discuss here about the meaning of \(\sqrt[n]{a}\).

The expression \(\sqrt[n]{a}\) means ‘nth rrot of a’. So, (\(\sqrt[n]{a}\))^n = a.

Also, (a^1/a)ⁿ = a ^{n × 1/n} = a¹ = a.

So, \(\sqrt[n]{a}\) = a^1/n.

Examples:

1. \(\sqrt[3]{8}\) = 8^1/3

          = (2³)^1/3

          = 2^{3 × 1/3}

          = 2¹

          = 2.

2. \(\sqrt[4]{9}\) = 9^1/4

          = (3²)¼   

          = 3^{2 × ¼}

          = 3^1/2

          = √3.

Note: 3^1/2 = \(\sqrt[2]{3}\). But \(\sqrt[2]{3}\) is also written as √3.

Solved examples on nth Root of a:

Express each of the following in the simplest form without radicals:

(i) \(\sqrt[4]{5^{2}}\)

(ii) \(\sqrt[n]{x^{m}}\)

(iii) \(\sqrt[3]{64^{-4}}\)

Solution:

(i) \(\sqrt[4]{5^{2}}\) = (5²)^1/4

             = 5^{2 × 1/4}

(ii) \(\sqrt[n]{x^{m}}\) = (x^m)^1/n

              = x^{m × 1/n}

              = x^m/n.

(iii) \(\sqrt[3]{64^{-4}}\) = (64^-4)^1/3

                 = 64^{-4 × 1/3}

                 = 64^-4/3

                 = (4³)^-4/3

                 = 4^3(-4/3)

                 = 4^-4

                 = \(\frac{1}{4 × 4 × 4 × 4}\)

                 = \(\frac{1}{256}\).

9th Grade Math

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