What you need to know

There are three rules of indices (or laws of indices) which you have to know and be able to apply to problems involving both numbers and algebra. For any numbers, x, m, and n, those three rules are

  • The multiplication law – when you multiply terms, you add the powers:

x^m\times x^n=x^{m+n}.

  • The division law – when you divide terms, you subtract the powers:

x^m\div x^n=\dfrac{x^m}{x^n}=x^{m-n}.

  • The power law – when you take a power of a term already with a power, the powers are multiplied:

\left(x^m\right)^n=x^{mn}.

These laws don’t appear from nowhere. For example, let’s consider what happens if we do

a^5\times a^3

If we write this out as individual multiplications, it becomes

\left(a\times a\times a\times a\times a\right)\times\left(a\times a\times a\right)

The left-hand bracket is a^5 and the right-hand bracket is a^3. How many times is a being multiplied by itself? Counting, we see that a is repeated 5+3=8 times in the expression. So, we get that

a^5\times a^3=a\times a\times a\times a\times a\times a\times a\times a=a^8

In mathematics, it’s important not to take things for granted. It’s a good exercise to try and figure out why the other two laws also make sense.

Example: Work out the value of \dfrac{3^4\times3^7}{3^8}. (No calculator)

If we tried to work out 3^8 without a calculator, it would take a long time. Fortunately, the laws of indices can make our lives a lot easier. Firstly, let’s consider the numerator. Applying the multiplication law, we get that

3^4\times3^7=3^{4+7}=3^{11}

So, the calculation becomes

\dfrac{3^{11}}{3^8}

This is a division, so applying the division law, we get

\dfrac{3^{11}}{3^8}=3^{11-8}=3^3

Now, the expression has become something that’s not too tricky to handle, and so the final answer is 3^3=9\times3=27.

Note: problems like this can only be simplified if the base is the same, i.e. we can do 3^4\times3^7=3^{11}, but there is no good way to simplify 3^4\times2^5 since their bases are different.

Example: Write 5p^2q^3\times3pq^4 in its simplest form.

To simplify this expression, we must recognise that it can be broken up into parts, i.e. we can write

5p^2q^3\times3pq^4=5\times p^2\times q^3\times3\times p\times q^4

Then, we can rearrange the terms of this multiplication to make it

5\times 3\times p^2\times p\times q^3\times q^4

Firstly, 5\times3=15. Then, the rest of the terms can be simplified using the multiplication law. Recalling that p=p^1, we get that

p^2\times p=p^3\,\text{ and }\,q^3\times q^4=q^7

Therefore, the expression simplifies to

15p^3q^7

This application of the multiplication law is used all the time when you expand brackets. With practice, you should be able to perform this process quite quickly.

This next example is a little trickier, so don’t worry if it isn’t so obvious at first.

Example: Write 2^{15}\times 8^{-4} as a power of 2, and hence evaluate the expression. (No calculator)

As mentioned earlier, the laws of indices only work if the things you’re multiplying/dividing have the same base, and these two things certainly don’t. However, with some subtle manipulation, we can make it so that they do.

The first part of the expression is a power of 2, whilst the second part is a power of 8. The key observation here is realising that 8 is actually a power of 2, specifically it is 2^3. This means we can write the second term in the expression as

8^{-4}=\left(2^3\right)^{-4}

Now, this is in the appropriate form for us to apply the power law. Doing so, we get

\left(2^3\right)^{-4}=2^{3\times(-4)}=2^{-12}

This is extremely helpful, because it is now also a power of two! So, because the whole expression can be written as

2^{15}\times2^{-12},

we can use the multiplication law. Hooray! So, we get

2^{15}\times2^{-12}=2^{15+(-12)}=2^3

Thus, we have written the expression as a power of 2. Evaluating the expression: 2^3, we get 2\times2\times2=8.

Note: if you’re taking the higher paper, there’s no reason why these types of questions can’t include fractional powers. That said, the ideas are exactly the same, you just have to be more careful with your calculations, since fractions make everything a bit more effort.

Example Questions

1) Work out the value of \dfrac{5^9}{5^{14} \div 5^7}. (No calculator)

Answer

There’s no way we can simplify the numerator, so let’s look to the denominator. To it, we can apply the division law. Doing so, we get the denominator to be

 

5^{14}\div5^7=5^{14-7}=5^7

 

Therefore, the expression becomes

 

\dfrac{5^9}{5^7}

 

So, we can apply the division law again. Doing so, we get

 

\dfrac{5^9}{5^7}=5^2

 

Thus, the value of the expression is 5^2=25.

2) Simplify the expression:

\dfrac{abc^2\times a^{3}c}{ab^2\times\left(c^2\right)^3}.

Answer

First, we will simplify the numerator. Breaking up the components, we can write it as

 

a\times a^3\times b\times c^2\times c

 

Applying the multiplication law, this becomes

 

a^{4}bc^{3}

 

Next, the denominator. Firstly, apply the power law to get that (c^2)^3=c^{2\times3}= c^6. Then, the denominator becomes

 

ab^{2}c^{6}

 

Thus, the expression becomes

 

\dfrac{a^{4}bc^{3}}{ab^{2}c^{6}}

 

With practice, you should get used to cancelling powers between the top and bottom of a fraction. That said, what we’re actually doing when we’re “cancelling down” is writing the expression like

 

\dfrac{a^4}{a}\times\dfrac{b}{b^2}\times\dfrac{c^3}{c^6}

 

Then, applying the division law 3 times, and simplifying, we get

 

a^{4-1}\times b^{1-2}\times c^{3-6}=a^{4}b^{-1}c^{-3}.

 

This completes the simplification of the expression. Note: if your answer was

 

\dfrac{a^4}{bc^3},

 

then this is also correct.

3) Write 9^5\times3^{-5} as a power of 3. (No calculator)

Answer

So, we can’t use any laws straight away since the terms don’t have the same base. However, if we recognise that 9=3^2, then we can write the first term as

 

\left(3^2\right)^5

 

Using the power law, we get

 

\left(3^2\right)^5=3^{2\times5}=3^{10}

 

Therefore, the whole expression becomes

 

3^{10}\times3^{-5}

 

Applying the multiplication law, this simplifies to

 

3^{10+(-5)}=3^5

 

Thus, we have written it as a power of 3.

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