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Indices 2

The Rules

a^m x aⁿ = a^m+n

a^m / aⁿ = a^m-n

(a^m)ⁿ = a^mn

More fractional indices

We have seen that a^1/n is the nth root of a. But what happens when the the numerator of the fractional index is not one?

9^3/2 = (9^1/2)³ using the third rule above.

9^3/2 = (3)³ = 27.

In general, a^m/n = (a^1/n)^m.

In other words find the nth root and then raise the answer to the power m.

Some examples

8^2/3 = (8^1/3)² = (2)² = 4

First find the cube root of 8: 8^1/3 = 2

Then square the answer: 2² = 4

32^0.6 = 32^3/5 = (32^1/5)³ = (2)³ = 8

First find the 5th root of 32: 32^1/5 = 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.