We've learnt about index laws already. Now we are going to look at how to apply these rules in questions with algebraic expressions. A power indicates how many times a number is multiplied by itself. For example, $m^3=m\times m\times m$m3=m×m×m. Let's look at how we can build on this knowledge.

Index Rules With Algebraic Expressions

Product rule

The product rule states: $a^m\times a^n=a^{m+n}$am×an=am+n

Let's look at how we derive this rule using an example. Say I wanted to simplify the expression $a^5\times a^3$a5×a3. In an expanded form, this would mean $a$a multiplied by itself 5 times multiplied by $a$a times itself 3 times:

So, you can see that $a$a is now multiplied by itself 8 times, which we can write as $a^8$a8 (which is the same as adding the powers, i.e., $a^{5+3}$a5+3).

Quotient Rule

The quotient rule states:$a^m\div a^n=a^{m-n}$am÷an=am−n

This is derived in a similar way to the product rule. Say we wanted to simplify the expression $a^6\div a^2$a6÷a2. In expanded form, we would write this as:

You can see I've taken out common factors to simplify the expression, leaving with an answer of $a^4$a4 (which is the same as subtracting the exponents, i.e., $a^{6-2}$a6−2)

Let's Simplify

Now let's look at how we would use these rules to simplify an algebraic expression.

Example

Simplify:$\frac{4h^3}{16h^5}$4h316h5

Think: If we wrote this in expanded form and simplified the fraction by taking out common factors, it would be:

So, if we were writing our simplified answer with a positive index, our answer would be $\frac{1}{4h^2}$14h2. We could also write this with a negative index as $\frac{1}{4}h^{-2}$14h−2.

More examples

Question 1

Simplify the following expression, giving your answer with positive powers of $u$u:

$\frac{36u^2}{4u^{10}}$36u24u10

Question 2

Simplify the following, giving your answer in index form: $\left(-5n^3\right)\times m^4\times\left(-5n^3\right)\times m^5$(−5n3)×m4×(−5n3)×m5.