Simplest form

The idea

Unlike terms cannot be combined, so an expression whose terms are all unlike is in its simplest form.

Pencils and erasers together

Over three days the shop earned 18c from pencils and 11d from erasers. Altogether: 18c + 11d.

Can 18c + 11d be written as one term?

• Yes: 29c
• Yes: 29cd
• No: c and d are different letter-numbers

Join until you cannot

Join the like terms. Then try joining terms that are unlike. See what happens.

When nothing more can be joined

Unlike terms cannot be joined. When all the terms of an expression are unlike, nothing more can be done: it is in its simplest form.

Unlike terms cannot be combined, so an expression whose terms are all unlike is in its simplest form.

• Expression | Simplest form?
• 18c + 11d | yes: c and d are unlike
• 5c + 3c + 10c | no: it becomes 18c
• 21p − 9q | yes: p and q are unlike

Notes

Unlike terms cannot be combined, so an expression whose terms are all unlike is in its simplest form: 18c + 11d.

Check yourself

Which expression is in simplest form?

Can 9x + 4y be made shorter?

Simplify 3m + 4n + 2m + n to ?m + ?n. How many n are there?

Answer: 5

3m + 2m = 5m and 4n + n = 5n, so 5m + 5n.

How many terms does the simplest form of 5p + 2q + p + 3q have?

• 6a + 2b — correct. Yes! a and b are different letter-numbers.
• 6a + 2a. These are like terms. They join into 8a.
• 6a + 2b + a. a is like 6a, so they join: 7a + 2b.
• No: x and y are unlike, so it is already in simplest form. — correct. Yes! Unlike terms cannot be combined.
• Yes: 13xy. Adding the number parts only works for like terms.
• Yes: 13x. The 4y is a different kind of term. It cannot become x.
• 4, because nothing can be joined. 5p and p are like terms, and so are 2q and 3q.
• 1, because everything joins. p and q are unlike. They cannot be joined.
• 2, because it is 6p + 5q — correct. Yes! 5p + p = 6p and 2q + 3q = 5q. p and q are unlike.