Laws of Indices

A power is a number multiplied by itself a number of times. 2^5 means 2\times2\times2\times2\times2, which is 32. The 2 is the base. The small raised 5 is the index, or power: it says how many 2s are multiplied together.

The index is not a multiplier: 2^5 is not 2\times5. Work it out by multiplying by 2 again and again: 2, 4, 8, 16, 32

The plural of index is indices. The laws of indices are a few short rules for multiplying and dividing powers of the same number without working them out first.

In the exam you are usually asked to write a calculation as a single power, such as “Give your answer as a power of 2”, to work out the value of a calculation with powers, or to find a missing index. These questions are usually worth 1 or 2 marks and are most often on the non-calculator paper, so learn the laws rather than relying on a calculator.

Look at 3^2\times3^4. That is two 3s multiplied together, then four more 3s: (3\times3)\times(3\times3\times3\times3). Altogether that is six 3s, so 3^2\times3^4=3^6

When you multiply powers of the same base, keep the base and add the indices:

a^m\times a^n=a^{m+n}

Here a is the base, and m and n are the indices.

Keep the base the same: 3^2\times3^4 is 3^6, not 9^6. And this law is only for multiplying. 3^2+3^4 is not 3^6: to add powers, work out each one first, 9+81=90

Look at 4^5\div4^2. Write it as a fraction and cancel: \frac{4\times4\times4\times4\times4}{4\times4}. The two 4s on the bottom cancel with two of the 4s on the top, leaving three 4s. So 4^5\div4^2=4^3

When you divide powers of the same base, keep the base and subtract the indices:

a^m\div a^n=a^{m-n}

Take the second index away from the first.

Subtract the indices; don't divide them. 4^6\div4^2 is 4^{6-2}=4^4, not 4^3

Look at (4^3)^2. The outside power 2 means two lots of 4^3 multiplied together: 4^3\times4^3=4^{3+3}=4^6

When a power is raised to another power, keep the base and multiply the indices:

(a^m)^n=a^{m\times n}

So (4^3)^2=4^{3\times2}=4^6, much more quickly.

Only multiply the indices when there is a power outside a bracket. Two powers written side by side with \times between them still have their indices added. If there is a calculation inside the bracket, do it first: (2^3\times2^2)^4=(2^5)^4=2^{20}

Divide a power by itself: 7^3\div7^3=7^{3-3}=7^0. But any number divided by itself is 1. So any number (except 0) to the power zero is 1:

a^0=1

So 10^0=1 and 25^0=1. The answer is 1, not 0 and not the base.

Now look at 2^3\div2^5. Subtracting the indices gives 2^{3-5}=2^{-2}. Working it out instead gives 8\div32=\frac{8}{32}=\frac14. So 2^{-2}=\frac14

A negative index means one over the positive power:

a^{-n}=\frac{1}{a^n}

So 2^{-2}=\frac{1}{2^2}=\frac14 and 10^{-2}=\frac{1}{10^2}=\frac{1}{100}=0.01. A negative index does not make the answer negative: it gives a small positive number.

The laws only work when the bases are the same. 2^3\times3^2 can't be written as one power, so work it out: 8\times9=72

Sometimes one base is a power of the other. These are worth knowing: 4=2^2, 8=2^3, 16=2^4, 9=3^2, 27=3^3, 25=5^2, 100=10^2 and 1000=10^3

Rewrite the base, then use a power of a power. For example, 16^2=(2^4)^2=2^{4\times2}=2^8

• Multiplying the bases or the indices. When you multiply powers, keep the base and add the indices: 7^2\times7^3=7^5, not 49^5 or 7^6
• Dividing the indices. When you divide powers, subtract the indices: 3^8\div3^2=3^6, not 3^4
• Thinking a power of zero is zero. Any number to the power zero is 1, so 4^0=1
• Thinking a negative index makes the answer negative. A negative index means one over the positive power: 4^{-2}=\frac{1}{16}, not -16 or -8
• Losing the base or slipping with minus signs. Write the base with every power and add negative numbers carefully: 2^{-4}\times2^9=2^5, because -4+9=5
• Stopping at a power when the question asks for the value. Work out the power as an ordinary number, so 3^4 becomes 81.