Fractional Indices

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

Fractional indices In general, x = x Also, we can write x as x . × m 1 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

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