Laws of indices
Home learning focus
Learn about the laws of indices.
This lesson includes:
- a learning summary
- one activity sheet
Using an index or power
An index, or power, is the small floating number that goes next to a number or letter. The plural of index is indices.
Indices show how many times a number or letter has been multiplied by itself.
Here is a number written in index form:
In this example, 2 is the base number and 4 is the index or power.
This is a short way of writing 2 × 2 × 2 × 2.
a³ (read as 'a cubed') is a short way of writing a × a × a. In other words, a has been multiplied by itself three times.
Simplify c³ × c².
To answer this question, you could write c³ and c² out in full:
c³ = c × c × c and c² = c × c.
Therefore, c³ × c² = c × c × c × c × c.
You can see that c has now been multiplied by itself 5 times.This means c³ × c² can be simplified to c⁵.
Another way to multiply indices is simply to add each index value to get a total.
In this example, 3 + 2 = 5. So, c³ × c² = c⁵
Note that this is is only true when the base number is the same. So, d³ × e² cannot be simplified because d and e are different values.
Simplify b⁵ ÷ b³.
One way to solve this is to consider that b⁵ ÷ b³ can be written as b⁵ over b³ (b⁵⁄ b³). Writing out the denominator (the bottom part of the fraction) and numerator (the top part of the fraction) in full gives:
There are common factors of b in the numerator and denominator and these can be cancelled out, which leaves b × b = b².
This means b⁵ ÷ b³ can be simplified to b².
A quicker way to divide indices with the same base number is to subtract the index values.
So, in this example 5-3=2 so b⁵ ÷ b³ = b²
Raising a power to a power
This means that k³ is to be squared, or multiplied by k³ again: (k³)² = k³ × k³
Add the powers together, so (k³)² = k³ × k³ = k⁶, so (k³)² can be simplified to k⁶.
A negative power, or negative index, is often referred to as a reciprocal.
The reciprocal of a number is 1 divided by that number. For example, the reciprocal of the number 4 is ¼.
The reciprocal of a fraction is that fraction turned upside down. For example, the reciprocal of ³⁄₄ is ⁴⁄₃.
Simplify d⁴ ÷ d⁵
Dividing indices means subtracting the index values, so:
d⁴ ÷ d⁵ = d⁴⁻⁵ = d⁻¹
This is an example of a negative index.
But you can write d⁴ ÷ d⁵ out and see that it is also equal to:
Cancelling common factors gives:
which shows that d⁴ ÷ d⁵ = ¹⁄d
So, d⁻¹ = ¹⁄d
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