Indices 2
The Rules
am x an = am+n
am / an = am-n
(am)n = amn
More fractional indices
We have seen that a1/n is the nth root of a. But what happens when the the numerator of the fractional index is not one?
93/2 = (91/2)3 using the third rule above.
93/2 = (3)3 = 27.
In general, am/n = (a1/n)m.
In other words find the nth root and then raise the answer to the power m.
Some examples
82/3 = (81/3)2 = (2)2 = 4
First find the cube root of 8: 81/3 = 2
Then square the answer: 22 = 4
320.6 = 323/5 = (321/5)3 = (2)3 = 8
First find the 5th root of 32: 321/5 = 2 (because 2x2x2x2x2 = 32)
Then cube the answer: 23 = 8
Raising a fraction to a power
(1/2)2 = (1/2) x (1/2) = 1/4
(3/4)3 = (3/4) x (3/4) x (3/4) = 27/64 = 33/43
We simply raise the top and the bottom of the fraction to the power.
(2/3)4 = 24/34 = 16/81
We raised the top to the power: 24 = 16
We raised the bottom to the power: 34 = 81
Raising a fraction to a negative power
We have seen that a negative power gives us one over an answer:
3-2 = 1 / 32 = 1/9
When you divide 1 by a fraction it flips the fraction upside down:
- 1 / (2/5) = 5/2
- 1 / (7/4) = 4/7
- 1 / (1/5) = 5/1 = 5
But we can use negative indices to give us one over something. So we can re-write the examples above:
- (2/5)-1 = 5/2
- (7/4)-1 = 4/7
- (1/5)-1 = 5
The index of -1 flips the fractions upside down. Other negative indices will also flip the fractions but they will change the numbers as well.
- (2/5)-2 = (5/2)2 = 25/4
- (7/4)-3 = (4/7)3 = 64/343
- (1/5)-4 = 54 = 625
Putting it all together
So, we've seen:
- Positive whole number indices
- Negative whole number indices
- An index of zero
- Fractional indices
- Fractions raised to powers
We should now be able to understand quite complicated examples.
(9/16)-3/2 = (16/9)3/2 = (4/3)3 = 64/27
First we flipped the fraction because of the negative power. 9/16 became 16/9.
Second we square rooted the top and bottom because the power had a denominator of 2.
Finally we cubed the top and bottom because the power had a numerator of 3.