1. Fractional Indices

2. Fractional indices Indices can also be fractional. x + = x × x = x1 =2
x × x = 1 2
x1 = x
But,
x × x = x
The square root of x.
So,
x = x 1 2
Similarly,
x × x × x = 1 3
x = 1 3
x1 = x
Discuss the fact that when we square the square root of a number we end up with the original number. Similarly, we get back to where we started when we cube the cube root of a number.
But,
x × x × x = x 3
The cube root of x.
So,
x = x 1 3

3. Fractional indices In general, x = x Also, we can write x as x . × m1 n n
Also, we can write x as x m n 1
× m
Using the rule that (xm)n = xmn, we can write
x × m = (x )m = (x)m 1 n
We can also write x as xm × . m n 1
x = (xm) = xm 1 n m×
In general,
x = xm
x = (x)m m n or

4. Index laws 1 xm × xn = x(m + n) x–1 = x 1 xm ÷ xn = x(m – n) x–n = xn
Here is a summary of the index laws.
x–1 = 1 x
xm × xn = x(m + n)
xm ÷ xn = x(m – n)
x–n = 1 xn
(xm)n = xmn
x = x 1 2
x1 = x
Discuss the general form of each result where x is any number and m and n are integers.
x = x 1 n
x0 = 1 (for x = 0)
x = xm or (x)m n m