Slideshow 6, Mathematics
Room 307, Mr. Sasaki
Multiplication and division
drill
Learn what a monomial is
Recall what happens when
we multiply something by
something else or itself
Recall what happens when
we divide something by
something else or itself
Practice multiplying and
dividing monomials
What is a monomial?
A monomial is a polynomial
that contains only one term
and a coefficient (even 1).
Is this a monomial?
2
4𝑥
Is this a monomial?
Yes!
3𝑦 + 2
Is this a monomial?
No!
2
2𝑎 𝑏
Is this a monomial?
Yes!
Basically, a monomial is a
polynomial without any addition
or subtraction operators.
When we multiply monomials,
we combine each monomial to
make one, single monomial. Let’s
have a look…
Example
Multiply 𝑎 by 𝑏.
Here are two monomials, a
𝑎 × 𝑏 = 𝑎𝑏
and b. In the simplest case,
we just write them together.
Algebraic expressions in their
simplest form do not contain
x or ÷ symbols, this is why it is
important to remove them.
Example
Simplify 𝑎 ÷ 𝑏.
𝑎
𝑎 ÷ 𝑏=
𝑏
When written as a fraction, this is thought of as
one number, one term. This is why this is
simplified in algebra.
Numbers follow rules that you should be familiar
with by now, let’s have a look.
Example
Simplify 4𝑥 2 × 2𝑦 .
2× 2𝑦
4𝑥
= 8 𝑥 2𝑦
When we have different unknowns (like 𝑥 and
𝑦), we just write them next to each other.
When we divide, the same applies however we
write the first unknown at the top and last at the
bottom.
Example
Simplify 4𝑥 2 ÷ 2𝑦 .
2 ÷ 2𝑦
4𝑥
= 2 𝑥2
𝑦
Try the short worksheet!
When we divide two different unknowns (like 𝑥
and 𝑦), we just write them the first above the
second.
Answers
2𝑥𝑦
𝑎𝑏𝑐
4𝑥3𝑦2
8𝑎𝑏𝑐
6𝑎𝑏
8𝑥3𝑦𝑧
30𝑎𝑏2
2𝑥
𝑦
2𝑎
𝑏
7𝑎2 𝑏
5𝑐𝑑 2
𝑎2 𝑐 2
+
𝑏
𝑑
You can see here that it’s
much easier to read this as a
set of fractions!
When terms are the same as each other, they
combine as one term in a different way.
Example
Simplify 𝑎 × 𝑎.
𝑥 𝑎x 𝑥 𝑏 = 𝑥 𝑎+𝑏
In algebra, we never
2write 𝑎𝑎.
1 = 𝑎. We
Also,
𝑎
Simplify 𝑎2 × 𝑎3. never write 𝑎1.
𝑎 × 𝑎= 𝑎
2
𝑎
×
3
𝑎
=(𝑎 × 𝑎) × (𝑎 × 𝑎 × 𝑎)
=𝑎×𝑎×𝑎×𝑎×𝑎
5
=𝑎
Division works in the opposite way.
Example
Simplify 𝑎4 ÷ 𝑎2.
𝑥 𝑎÷ 𝑥 𝑏 = 𝑥 𝑎−𝑏
4
𝑎
÷
2
𝑎
=𝑎 × 𝑎 × 𝑎 × 𝑎
=
2
𝑎
𝑎×𝑎
So when we multiply powers of the same term,
the power is added and when we divide, the
power is subtracted. Try to remember this!
Let’s try some general examples.
𝑎
0
Example
Note: 𝑎 means = 1.
𝑎
Simplify 4𝑎4𝑏 ÷ 2𝑎2.
We never write 𝑎0.
4
4𝑎 𝑏
÷
2
= 2𝑎 𝑏
2
2𝑎
Example
Simplify 4𝑎4𝑏 × 2𝑎2.
4
4𝑎 𝑏 ×
6
= 8𝑎 𝑏
2
2𝑎
Try the last worksheet!
Answers
5𝑎𝑏2
𝑎7
7𝑎2𝑏
21𝑎7
𝑥9
2𝑎7𝑏2
9𝑦4
𝑥5
4𝑎8
30𝑏3
1
𝑎2
−2 or 2
2𝑦
𝑦2
3𝑏5𝑎3