Meaning of Numbers with Rational Indices

Meaning of Numbers with Rational Indices:

I. Meaning of the numbers when an index is a rational number of the form `1/n`.

Let us see the meaning of indices in the form of rational numbers such as `1/2, 1/3, 1/5, ......, 1/n`.

To show the square of a number, the index is written as 2, and to show the square root of a number, the index is written as `1/2`.

For example, the square root of 25, is written as `sqrt(25)  "using the radical sign"   sqrt.`

Using the index, it is expressed as `25^(1/2) ∴ sqrt(25) = 25^(1/2)`

In general, the square of a can be written as a² and the square root of a is written as `root(2)(a) or sqrt(a) or a^(1/2).`

Similarly, a cube of a is written as a³, and cube root of a is written as `root(3)(a) or a^(1/3).`

For example, 4³ = 4 × 4 × 4 = 64.

∴ cube root of 64 can be written as `root(3)(64) or (64)^(1/3). "Note that", 64^(1/3) = 4`

3 × 3 × 3 × 3 × 3 = 3⁵ = 243. That is the 5^th power of 3 is 243.

Conversely, 5^th root of 243 is expressed as `(243)^(1/5) or root(5)(243). "Hence", (243)^(1/5) = 3.`

In the general n^th root of a is expressed as `a^(1/n).`

II. The meaning of numbers, having an index in the rational form `m/n`.

We know that 8² = 64,

Cube root at 64 is = `(64)^(1/3) = (8^2)^(1/3) = 4`

∴ cube root of the square of 8 is 4.      ..........(I)

Similarly, cube root of `8 = 8^(1/3) = 2`

∴ square of the cube root of 8 is `(8^(1/3))^2 = 2^2 = 4` ..........(II)  

From (I) and (II)

cube root of the square of 8 = square of cube root of 8. Using indices, `(8^2)^(1/3) = (8^(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)^(1/3) = (8^(1/3))^2 = 8^(2/3).`
From this, we get two meanings of the number `8^(2/3).`

(1) `8^(2/3) = (8^2)^(1/3)` i. e. cube root of the square of 8.

(2) `8^(2/3) = (8^(1/3))^2` i. e. square of cube root of 8. 

Similarly,

`27^(4/5) = (27^4)^(1/5)` means 'fifth root of fourth power of 27'.

and `(27)^(4/5) = (27^(1/5))^4` means 'fourth power of fifth root of 27'.

Generally, we can express two meanings of the number `a^(m/n)`.

`a^(m/n) = (a^m)^(1/n)` means 'n^th root of m^th power of a'.

`a^(m/n) = (a^(1/n))^m` means 'm^th power of n^th root of a'.