__Objective__

At the end of the lesson, the students should be able to simplify algebraic expressions involving multiplication and division.

In performing multiplication and division of algebraic expressions it is important to first multiply the number, then the letters. When you have like terms in the expression, you simplify them.

** ****Multiplication and Division of Algebraic Expressions by a Number (constant).**

In multiplication or division of algebraic expressions by a number, you don’t have to worry about whether the variables are the same or not, all you have to do is multiply or divide every expression.

__Example 1__

Simplify:

a) 2×t b) m×2 c) 2t×5

__Solution __

a) 2 ×t=2t b) m ×2=2m c) 2t×5=10t

**The convention is to write the number first then the letters. **

In algebraic expression, division can be expressed as Therefore when dividing algebraic expressions, it is written in the form even when it is given as

or a ÷ b.

This is done to make calculation easier.

__Example 1__

Simplify the following:

a) 2x ÷ 2 b) 4m ÷ 3 c) (3x+5) ÷ 5

__Solution __

**Multiplication and Division of Algebraic Expressions by a Number and a Letter**

It is important to first group the numbers with the same letter together and apply the basic operating rules of indices. After multiplication, the letters are written in alphabetical order.

To simplify with exponents, it is often simpler to work directly from the definition and meaning of exponents. For instance:

**Simplify ***x*^{6}** **×** x^{5}**

Just thought about what exponents mean. The ” *x*^{6} ”

means “six copies of *x* multiplied together”, and the ” *x*^{5} ” means “five copies of *x* multiplied together”. So if I multiply those two expressions together, I will get eleven copies of *x* multiplied together. That is:

*x*^{6} × *x*^{5} = (*x*^{6})(*x*^{5})

= (** xxxxxx**)(

=

=

Thus: *x*^{6} **×** *x*^{5} **=**x^{(6+5)}= *x*^{11 }

**Simplify **

__Example 1__

Simplify the following:

- a) 2a
^{2 }x 5a^{3}b) y x 4y x 3y c) 3c^{2}×3c^{3 }

__Solution 1__

a) 2**a**^{2}×5**a**^{3 }**Multiply the same variable using powers.**

= (2×5)**a**^{(2+3)} **The indices are added together**

=10**a**^{5 }

b) **y** x 4**y** x 3**y**

= (1×4×3)**y**^{ (1+1+1)}

=12y^{3}

**Note** that, y is the same as 1y.

c) 3c^{2}×3c^{3}

=3×3c^{ (2+3)}

=9c^{5}

d) (8d^{3}×2d^{5}) ÷4d^{2} **Deal with the bracket first.**

__Example 2__

a) 2*x *×3*y *× 5*z *b) –3x × 2y

__Solution__

a) 2*x ***×**3*y ***×** 5*z*

2*x × *3*y × *5*z* ** multiply the numbers, and rewrite the letters**

= (2**×**3**×** 5)*xyz*

= **30 xyz.**

b) –3x × 2y

–3x × 2y **multiple the number and multiple the letters**

= (–3×2)x × y

= **–6xy**

Therefore –3x × 2y=–6xy

__Example 3__

Simplify the following:

a) **2y x 6e x 3zy ** b) **3xy ^{2 }× 4x^{3}y**

__Solution __

a) 2**y** x 6**e** x 3**zy**

= (2×6×3)**y **× **y **× **e **× **z ** always arrange letters in alphabetical order

= 36**ey**^{2}**z**

^{ }

b) 3**xy**^{2 }× 4**x**^{3}**y**^{4}

=3×4**x**×**x**^{3}×**y**^{2}×**y**^{4}

=(3×4) **x**^{(1+3)}**y**^{(2+4)} **remember x=x ^{1}**

=12**x ^{4}y^{6}**

Therefore 3xy^{2 }× 4x^{3}y^{4}=**12x ^{4}y^{6}**

**When you have the same variable appearing more than once, you raise the variable to the number of times the variable appears.**

** Example 1**

Simplify the following:

** Solution**

Don’t forget that the “7” and the “3” are just numbers. Since 3 doesn’t go evenly into 7, we can’t cancel the numbers. The ^{7}/_{3} stays as it is. For the variables, we have three extra copies of *x* on top, so the answer is**: **

Numbers cancel out numbers and like variables also cancel out. For the above example, s^{2} cancels out s^{2} out of s^{3} leaving one “s”.

Like terms can be cancelled out, one of* x* the will be cancelled from the* x*^{2} leaving only one *x*. The answer therefore becomes

Login

Accessing this unit requires a login, please enter your credentials below!