Simplifying expressions

Collecting like terms

Collecting like terms means to simplify terms in expressions in which the variables are the same. In the expression 5a + 2b + 3a - 6b, the terms 5a and + 3a are like terms, as are 2b and -6b.

Example 1

Simplify b + b + b + b.

Adding the four like terms together gives 4b.

Example 2

Simplify 5m + 3m - 2m.

In this expression, all the terms are like terms as the variable in each term is m. Simplify the expression in order:

5m + 3m = 8m

8m - 2m = 6m

Question

Simplify 9c -7d + c + 3d + 5.

This expression contains three types of terms: the terms that contain c's, terms that contain d's and terms that are numbers alone.

To simplify this expression, collect the like terms.

9c -7d + c + 3d + 5

9c + c = 10c

-7d + 3d = -4d

+5

This gives 10c -4d + 5.

Question

Simplify 2p^2 + 3p + p^2.

This expression contains two types of different terms, those that contain p^2 and those that contain p. p^2 and p are not like terms because although they contain the same letter, the letters do not have the same power.

2p^2 + 3p + p^2

2p^2 + p^2 = 3p^2

+ 3p

Putting the simplified terms together gives 3p^2 + 3p.

curriculum-key-fact
If there is only one of the letter, like in the answer to 5p - 4p, do not write the 1. A letter on its own means that there is only one of them, for example, 5p - 4p = p.

Using letters and numbers

Algebraic expressions can be added and subtracted by collecting like terms, but expressions can also be multiplied and divided.

Example 1

Simplify a \times a.

Multiplying a number or letter by itself is called squaring. This means a \times a = a^2 (read as 'a squared'). In a^2, the 2 is known as the index number or power. Powers tell us how many times a number or letter has been multiplied by itself.

Example 2

Simplify b \times b \times b.

In this example, b is being multiplied by itself three times. The power of b will be three, so, b \times b \times b = b^3.

Question

Simplify 3d \times 5d.

Multiply the numbers first. This gives 3 \times 5 = 15. Then multiply d \times d = d^2.

The final answer is 15d^2.

Question

Simplify 16e^2 \div 2e.

Dividing the variable and the numbers separately gives 16 \div 2 = 8 and e^2 \div e = e, so 16e^2 \div 2e simplified is 8e.