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 \(8e\).