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# Algebraic Multiplication and Division Using the Index Laws

Lesson

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=amn

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., $a^{6-2}$a62)

## 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}$14h2.

#### 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.

##### Question 3

Convert to fraction form and simplify:

$\left(\left(-4v^8\right)\times u^4\right)\div\left(\left(-2v^8\right)\times u^{10}\right)$((4v8)×u4)÷​((2v8)×u10)