Topics to be learn : Meaning of numbers with rational indices Cube and Cube Root

Introduction :

The product 2 × 2 × 2 × 2 × 2 = 2⁵, in which 2 is the base, 5 is the index and 2⁵ is the index form of the number.

Laws of indices : If m and n are integers, then

(i) a^m × aⁿ = a^m+n

• Ex. 3⁵ × 3² = 3⁷

(ii) a^m ÷ aⁿ = a^m-n

• Ex. 3⁷ ÷ 3⁹ = 3^-2

(iii) (a × b)^m = a^m × b^m

• Ex. (5 × 7)² = 5² × 7²

(iv) a⁰ = 1

• Ex. 5⁰ = 1

(v) a^-m = \(\frac{1}{a^m}\)

• Ex. 5^-3 = \(\frac{1}{5^3}\)

(vi) (a^m)ⁿ = a^mn

• Ex. (3⁴)⁵ = a²⁰

(vii) \((\frac{a}{b})^m\) = \(\frac{a^m}{b^m}\)

• Ex. \((\frac{5}{7})^3\) = \(\frac{5^m}{7^m}\)

(viii) \((\frac{a}{b})^{-m}\) = \(\frac{b^m}{a^m}\)

• Ex. \((\frac{5}{7})^{-3}\) = \((\frac{7}{5})^3\)

Meaning of numbers with rational indices :

(I) Meaning of the numbers when the index is a rational number of the form

(i) To show the square of a number, we write the index 2.
Example: square of a = a²

To show the square root of a number, we write the index \(\frac{1}{2}\)  or use the \(\sqrt{\,\,\,\,}\) sign.
Example: square root of 25, is written as \(\sqrt{25}\) = \(25^{\frac{1}{2}}\)

In general:

• Square of a = a²
• Square root of a = \(\sqrt{a}\) = \(a^{\frac{1}{2}}\)

(ii) Similarly, to show the cube of a number, we write the index 3.
Example: cube of a = a³

The cube root of a number is written using the cube root sign or index \(\frac{1}{3}\) 

Example: Cube root of a = \(\sqrt[3]{a}\) = \(a^{\frac{1}{3}}\)

For example, 4³ = 4 × 4 × 4 = 64.

∴ cube root of 64 can be written as \(\sqrt[3]{64}\) or \(64^{\frac{1}{3}}\)

Note that, \(64^{\frac{1}{3}}\) = 4

(iii) Similarly, nth root of a is expressed as \(a^{\frac{1}{n}}\)

For example, 3 × 3 × 3 × 3 × 3 = 3⁵ = 243. i.e. 5th power of 3 is 243.

Conversely, 5th root of 243 is expressed as \(\sqrt[5]{243}\) or \(243^{\frac{1}{5}}\)  Hence, \(243^{\frac{1}{5}}\) = 3

∴ If \(a^{\frac{1}{m}}\) = x then x^m = a.

(II) The meaning of numbers, having index in the rational form \(\frac{m}{n}\)

We know that 8² = 64,

Cube root at 64 is = \(64^{\frac{1}{3}}\) = \((8^2)^{\frac{1}{3}}\) = 4

∴ cube root of square of 8 is 4                         …..(i)

Similarly, cube root of 8 = \(8^{\frac{1}{3}}\) = 2

∴ square of cube root of 8 is = 2² = 4       ……(ii)

From (i) and (ii) cube root of square of 8 = square of cube root of 8.

Using indices,  \((8^2)^{\frac{1}{3}}\) = \((8^{\frac{1}{3}})^2\)

The rules for rational indices are the same as those for integral indices

∴ using the rule (a^m)ⁿ = a^mn, we get  \((8^2)^{\frac{1}{3}}\) = \((8^{\frac{1}{3}})^2\) = \(8^{\frac{2}{3}}\).

In general, we can express two meanings of the number \(a^{\frac{m}{n}}\)

\(a^{\frac{m}{n}}\) = \((a^m)^{\frac{1}{n}}\) means ‘n^th root of m^th power of a’.

\(a^{\frac{m}{n}}\) = \((a^{\frac{1}{n}})^m\) means ‘m^th power of n^th root of a’.

Cube and Cube Root :

Cube :

If a number is written 3 times and multiplied, then the product is called the cube of that number.

e.g. 5 x 5 x 5 = 5³ = 125. So 125 is the cube of 5.

Ex. 1. The cube of (-4) = (-4)³ = (-4) × (-4) × (-4) = - 64.

Ex. 2. The cube of \(\frac{5}{6}\) = \((\frac{5}{6})^{\frac{1}{3}}\) = \(\frac{5}{6}\) × \(\frac{5}{6}\) × \(\frac{5}{6}\) = \(\frac{125}{216}\)

Ex. 3. The cube of (1.2) = (1.2)³ = 1.2 × 1.2 × 1.2 = 1.728

Ex. 4. The cube of 0.03 = (0.03) × (0.03) × (0.03) = 0.000027.

Remember :

(i) The cube of a positive number is positive.

(ii) The cube of a negative number is negative.

(iii) The cube of a decimal fraction : Observe the number of decimal places in the number. The number of decimal places in the cube of the number = 3 x the decimal places in the number.

To find the cube root :

To find the cube root of a given number, find its prime factors.

Ex. 1 To find the cube root of 216.

First find the prime factor of 216.

Factor of 216 = 2 × 2 × 2 × 3 × 3 × 3

Each of the factors 3 and 2, appears thrice. So let us group them as given below,

216 = (3 × 2) × (3 × 2) × (3 × 2) = (3 × 2)³ = 6³

∴ \(\sqrt[3]{216}\) = 6 that is \(216^{\frac{1}{3}}\) = 6

Ex. 2. Find \(\sqrt[3]{0.125}\)

Solution :

\(\sqrt[3]{0.125}\) = \(\sqrt[3]{\frac{125}{1000}}\)

= \(\frac{\sqrt[3]{125}}{\sqrt[3]{1000}}\)      ….. \((\frac{a}{b})^m=\frac{a^m}{b^m}\)

= \(\frac{\sqrt[3]{5^3}}{\sqrt[3]{10^3}}\) = \(\frac{5}{10}\) = 0.5