Lesson 3: Multiplying terms (with indices)


Multiplying terms gets a bit more challenging when indices are involved. But if you take your time and break it up into manageable steps, it becomes much easier. This lesson has been split into three sections to help you.


Let's start by getting to know index notation. You should notice a pattern in this list.


\(a\times a~=~a^2\)

\(a\times a\times a~=~a^3\)

\(a\times a\times a\times a~=~a^4\)

\(a\times a\times a\times a\times a~=~a^5\)

\(a\times a\times a\times a\times a\times a~=~a^6\)

\(a\times a\times a\times a\times a\times a\times a~=~a^7\)

Index notation (or index form) is just shorthand - that's all! The little number (called the index, or power) tells us how many \(a\)'s are being multiplied together.

How to speak in index form!

\(x^2\)  is pronounced \(x\) squared
\(x^3\)  is pronounced \(x\) cubed
\(x^4\)  is pronounced \(x\) to the power of 4
\(x^5\)  is pronounced \(x\) to the power of 5, etc.


1) Simplify  \(x\times x\times x\times x\)  (write it in index form, or using index notation).

       Answer: \(x^4\)

2) What does \(y^5\) represent? (what does it mean?)

       Answer: \(y\times y\times y\times y\times y\) 

3) What does \(z^1\) represent?

       Answer: \(z\)  (raising something to the power of 1 doesn't change it!)

       An index of 1 often catches people out!

Practise to master


Simplify each of these expressions using your knowledge of indices.

01)  \(c\times c\)


02)  \(n\times n\times n\times n\)


03)  \(x\times x\times x\)


04)  \(y\times y\times y\times y\times y\)


05)  \(b\times b\times b\times b\times b\times b\times b\times b\times b\)


06)  \(r\times r\times r\times r\times r\times r\)


Change the form of each of these expressions.

07)  \(m^5\)

\(m\times m\times m\times m\times m\)

08)  \(d^1\)


09)  \(g\times g\times g\)


10)  \(k^1\)


11)  \(p\times p\)


12)  \(w^{12}\)

\(w\times w\times w\times w\times w\times w\times w\times w\times w\times w\times w\times w\)


If any terms are already raised to a power (greater than 1), unsimplify these first. This opens things up a bit and helps us to understand what we need to do.

Example 1

Simplify  \(a^3\times a\times a^2\)

Brackets make it easier for you to follow:  \((a^3)\times (a)\times (a^2)\)

Unsimplify the terms in brackets:  \((a\times a\times a)\times (a)\times (a\times a)\)

Get rid of the brackets (they're in the way now):  \(a\times a\times a\times a\times a\times a~=~a^6\)

An even more impressive method

Surely it was obvious at the start that we were going to end up with 3 \(a\)'s, 1 \(a\) and 2 \(a\)'s multiplied together, which is a total of 6 \(a\)'s multiplied together, which is \(a^6\) ?

Of course, you just need to add the powers (3 + 1 + 2 = 6)!

Careful, though, \(a\) without an index is the same as \(a^1\)  (see Section A). It's a really common mistake to forget this invisible 1 when you're adding the powers together!

This method or rule is written formally like this: \(x^a\times x^b~=~ x^{a+b}\)

Example 2

Simplify  \(y^2\times y^3\times y^2\times y\)

Answer:  \(y^8\)  since 2 + 3 + 2 + 1 = 8 (I didn't forget the 1 ✌).

Practise to master


Simplify these expressions using your knowledge of indices.

01)  \(y^2\times y\)


02)  \(a^5\times a\)


03)  \(m^4\times m^2\)


04)  \(x^2\times x^3\)


05)  \(e^2\times e\times e^2\)


06)  \(x\times x^5\times x^3\)


07)  \(y^3\times y\times y^2\)


08)  \(b^4\times b^3\times b\)


09)  \(x\times x^2\times x^3\times x\)


10)  \(y^5\times y\times y\times y^2\)


11)  \(g^7\times g^2\times g\times g^3\)


12)  \(s^4\times s\times s^2\times s^3\times s\)



This third section combines everything you've learned on this course so far. Only attempt these questions if you've understood it all.

Example 1

Simplify  \(b\times c\times b\times a\)

Arrange the letters in alphabetical order:  \(a\times b\times b\times c\)

Simplify the  \(b\)  terms using your knowledge of indices:  \(a\times b^2\times c\)

Get rid of the multiplication signs:  \(ab^2c\)

Example 2

Simplify  \(2b\times 3b\times b\)

Take it apart:  \(2\times b\times 3\times b\times b\)

Rearrange:  \(2\times 3\times b\times b\times b\)

Multiply the numbers and use your knowledge of indices:  \(6\times b^3\)

Get rid of the remaining multiplication sign:  \(6b^3\)

Example 3

Simplify  \(3a^2\times 4ab\times b^3\)

Take it apart:  \(3\times a^2\times 4\times a\times b\times b^3\)

Rearrange:  \(3\times 4\times a^2\times a\times b\times b^3\)

Multiply the numbers and use your knowledge of indices:  \(12\times a^3\times b^4\)

Get rid of the remaining multiplication signs:  \(12a^3b^4\)

A note about workings - you don't need any! As you become more confident with this, you will look at the question and just write down the answer.

Practise to master


Simplify the following expressions.

01)  \(x\times y\times x\)


02)  \(y\times x\times y\)


03)  \(b\times a\times 4\times a\times a\)


04)  \(2\times c\times 7\times b\times c\)


05)  \(3a^2\times 2a^3\)


06)  \(2x^5\times 5x\)


07)  \(4y^3\times y^2\times 3y^2\)


08)  \(2b^4\times 2b\times b^2\)


09)  \(3g^2\times 2gh\)


10)  \(2pq^2\times 3p^3\)


11)  \(5x^2\times 2xy\times x^3y^2\)


12)  \(2y^2z\times yz^3\times 3y^3\)


13)  \(st^2\times 3rst\times 2r^2st\)


14)  \(x^2y\times 3xyz^3\times 2xy^2z\)


15)  \(4cd^2\times 3c^3de\times 2cde^2\)


16)  \(3pq^2\times 4p^3qr\times pqr^2\)