Lesson 2.1 · 25 min

Simplifying and the Laws of Indices

Multiplying and dividing algebraic terms with the index laws, and collecting like terms.

In this lesson you will learn to…

Multiply and divide terms such as $3 x^{3} \times 4 x^{2}$ and $12 p^{3} \div 2 p^{2}$
Raise a term to a power, e.g. $(2 x^{2})^{4}$
Cancel common letters in an algebraic fraction
Identify like terms and collect them to simplify an expression

Why this matters

Almost every algebra question you will meet — solving equations, rearranging formulae, working with quadratics — starts by tidying up an expression. The faster and more accurately you can simplify, the less room there is for mistakes later.

Multiplication signs go missing

In algebra we leave out the $\times$ signs:

$12 \times x = 12x \qquad 3 \times p \times q = 3pq \qquad 2 \times 5m \times 10n = 100mn$

Numbers are written first, then letters, usually in alphabetical order. When you are cancelling, it can help to write the long form back in:

$\frac{3x^3y^2z^2}{x^2y^3z} = \frac{3 \times x \times x \times x \times y \times y \times z \times z}{x \times x \times y \times y \times y \times z}$

and then cross out matching letters top and bottom.

Why the index laws work

$a^5 \times a^3 = (a \times a \times a \times a \times a) \times (a \times a \times a) = a^8$

Count the $a$ s: $5 + 3 = 8$ . Dividing cancels them instead:

$\frac{a^5}{a^3} = \frac{a \times a \times \cancel{a} \times \cancel{a} \times \cancel{a}}{\cancel{a} \times \cancel{a} \times \cancel{a}} = a^2$

Like terms

An expression is a collection of terms separated by $+$ and $-$ signs. You can only add or subtract like terms — terms with the same letters to the same powers.

Like terms	Not like terms
$2a$ and $5a$	$2a$ and $2a^2$
$ab^2$ and $10ab^2$	$ab^2$ and $a^2b$
$xy^3$ and $6xy^3$	$xy$ and $xz$

Order does not matter in multiplication. $ab$ and $ba$ are the same term, so $3ab + 2ba = 5ab$ . But $a^2b$ and $ab^2$ are genuinely different.

Watch the signs

The sign in front of a term belongs to that term. In $3a + 4b - 2a + 5b$ the $-$ is attached to the $2a$ , so the $a$ terms give $3a - 2a = a$ and the answer is $a + 9b$ .

Key rules and formulas

The index laws, with letters
$a^{m} \times a^{n} = a^{m + n} a^{m} \div a^{n} = a^{m - n} (a^{m})^{n} = a^{mn}$
Exactly the same laws as with numbers. They only combine powers of the same letter.
Numbers and letters separately
Multiply (or divide) the numbers, then deal with each letter on its own. $3 x^{3} \times 4 x^{2} = (3 \times 4) \times x^{3 + 2} = 12 x^{5}$ .
A bracket to a power hits everything inside
$(2 x^{2})^{4} = 2^{4} \times x^{2 \times 4} = 16 x^{8}$
The most common slip is $(2 x^{2})^{4} = 2 x^{8}$ — forgetting that the 2 is also raised to the power 4.
Like terms
Like terms have exactly the same letters raised to the same powers. $3 ab$ and $5 ab$ are like terms; $a b^{2}$ and $a^{2} b$ are not. Only like terms can be added or subtracted.

Worked examples

Worked example 1

Multiplying and dividing terms

Simplify: a

3 x^{3} \times 4 x^{2}

y^{2} \div y^{10}

(x^{5})^{3}

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Cancelling an algebraic fraction

Simplify

\frac{3 x ^{3} y ^{2} z ^{2}}{x ^{2} y ^{3} z}

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Collecting like terms

Simplify a

x y + 2 ab - 5 x y + 10 ab

2 n^{2} + m^{2} - n^{2} + 2 m^{2}

Try it yourself first, then reveal the solution one step at a time.

Practice: simplifying and indices

10 questions

Type expressions the way you would write them — 12x^5 for

12 x^{5}

, a/b for a fraction. Any equivalent form is marked correct.

Question 1 of 10Algebraic answerFoundation
Simplify $2 a^{3} \times 4 a^{4}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 2 of 10Algebraic answer
Simplify $\frac{14 y ^{10}}{7 y ^{5}}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 3 of 10Multiple choice
Simplify $(2 x^{2})^{4}$ .
Question 4 of 10Match the pairs
Match each expression to its simplified form.
Pick an answer, then choose the row it belongs to. You can also drag it.
- $x^{2} \times x^{3}$
- $x^{8} \div x^{4}$
- $(x^{3})^{2}$
- $x^{4} \times x^{2} \times x$
- $x^{5} \div x^{8}$
Question 5 of 10Algebraic answer
Simplify $\frac{3 x ^{3} y ^{2} z ^{2}}{x ^{2} y ^{3} z}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 6 of 10Choose all that apply
Jamil's homework is below. Select every line he got wrong.
Select every correct answer — there may be more than one.
$(3 p)^{3} = 9 p^{9}$ $4 t^{4} \times 4 t^{4} = 16 t^{8}$ $8 y^{0} = 1$ $8 m^{8} \div 2 m^{2} = 4 m^{4}$
Question 7 of 10Algebraic answer
Collect like terms: $5 x y + 2 z - 3 x y - 5 z$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 8 of 10True or false
$a b^{2}$ and $a^{2} b$ are like terms.
Question 9 of 10Choose all that apply
Hannah simplified these. Select every answer that is correct.
Select every correct answer — there may be more than one.
$3 a b^{2} + 5 b a^{2} + b^{2} a = 9 a b^{2}$ $3 pq - 8 pq + 4 pq = - pq$ $(3 x)^{2} - 3 x^{2} = 0$ $7 p^{2} - 6 p^{2} = p^{2}$
Question 10 of 10Algebraic answerChallenge
Simplify $\frac{a ^{2} b c ^{3}}{a ^{5} b ^{2}} \times \frac{a b ^{3} c ^{2}}{b c ^{4}}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.

In this lesson you will learn to…

Why this matters

Multiplication signs go missing

Why the index laws work

Like terms

Watch the signs

Key rules and formulas

The index laws, with letters

Numbers and letters separately

A bracket to a power hits everything inside

Like terms

Worked examples

Practice: simplifying and indices