Algebra

Ontario 09 Academic (MPM1D)

Multiply and divide algebraic terms with indices

Lesson

We've already looked at how to simplify and substitute values into algebraic expressions. Now we are going to look at how to add subtract, multiply and divide algebraic fractions. The processes we follow with algebraic fractions are very similar to the processes we use with fractions that only contain numbers. Let's check it out.

Do you remember that when we are adding or subtracting fractions, we need to have common denominators? Well, the same goes for algebraic fractions. We need to find a common factor between the fractions (remember you can always multiply the denominators together to find a common factor).

1. Find a common denominator

2. Multiply the numerators by the same number as the denominators to keep the fractions equivalent

3. Add the numerators

4. Simplify the fraction if possible.

We've already looked at how to multiply and divide fractions, and the processes are the same for algebraic fractions. We just need to consider any algebraic factors when we are simplifying our answers.

Simplify $\frac{8x}{12}+\frac{10x}{12}$8`x`12+10`x`12

Simplify the following: $\frac{-2x}{11}\div\frac{7y}{5}$−2`x`11÷7`y`5

Simplify the following: $\frac{-3x}{7}\times\frac{12y}{7}$−3`x`7×12`y`7

We've already seen about exponent 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, $m^3=m\times m\times m$`m`3=`m`×`m`×`m`. Let's look at how we can build on this knowledge.

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

Let's look at how we derive this rule using an example. Say I wanted to simplify the expression $a^5\times a^3$`a`5×`a`3. 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 $a^8$`a`8 (which is the same as adding the powers, i.e., $a^{5+3}$`a`5+3).

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

This is derived in a similar way to the product rule. Say we wanted to simplify the expression $a^6\div a^2$`a`6÷`a`2. 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 $a^4$`a`4 (which is the same as subtracting the exponents, i.e., $a^{6-2}$`a`6−2)

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

**Simplify:** $\frac{4h^3}{16h^5}$4`h`316`h`5

**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 exponent, our answer would be $\frac{1}{4h^2}$14`h`2. We could also write this with a negative exponent as $\frac{1}{4}h^{-2}$14`h`−2.

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

$\frac{36u^2}{4u^{10}}$36`u`24`u`10

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

Convert to fraction form and simplify:

$\left(\left(-4v^8\right)\times u^4\right)\div\left(\left(-2v^8\right)\times u^{10}\right)$((−4`v`8)×`u`4)÷((−2`v`8)×`u`10)

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