U
B
N
H
S
Lord teach me to number my days
And graph them according to your
ways
Trusting you to base me in my plan
To complement your perfect diagram
Subtract the points you do not want
from me
But add the values you have set at me
Divide the dividend I possess
accordingly
So I can multiply them systematically.
Draw the lines I have to follow
Guide me properly with your arrow
Because sometimes I tend to be
irrational
Yet all the while you want me to be
rational.
Well I learn that life is like a slope
With its ascents and descents that I
must cope
Going through such a wonderful
formula
Is just like solving problems in
Mathematics 9.
 Amen
Let’s Recall
64
8
±8
-8
Let’s Recall
2
12𝑥
2x 3
4x 3
x 12
Let’s Recall
5∙
1
5
25
=5
Let’s Recall
3
4
6
16
4∙
3
=4
Let’s Recall
𝑥−1 𝑥+1
x2 - 1
x2 -2x -1
x2 + 1
Let’s Recall
3𝑥 − 2 3𝑥 + 2
6x2 - 4
9x2 - 4
9x2 -12x - 4
Let’s Recall
2+ 3 2− 3
5
-1
1
The Quotient Law of Radicals
If a and b represent a nonnegative real
numbers and n is a positive integer,
Thus, the nth root of the quotient of two
nonnegative number is equal to the quotient of
their nth root.
𝒏
𝒂
=
𝒃
𝒏
𝒂
𝒏
𝒃
75
36
=
75
36
=
5 3
6
25 ∙ 3
36
3
81
=
8
3
3
81
8
3
=
3
27.3
3
3 3
2
8
3
=
33 ∙ 3
3
23
18𝑥 3
49𝑥𝑦 2
=
18𝑥 3
49𝑥𝑦 2
=
9 ∙ 2 ∙ 𝑥 2∙ 𝑥
49 ∙ 𝑥 ∙ 𝑦 2
3𝑥 2
7𝑦
3 ∙ 𝑥 2𝑥
=
7∙𝑦 𝑥
𝑤
=
6 6
3 2
2 3𝑚
18𝑦 2
36𝑥 8
Rationalizing the Denominator of a Radical Expression
A radical expression is not considered to be simplified
when its denominator contains radical. The process of
rewriting the radical such that the denominator will not
contain any radicals is called rationalizing the denominator.
1
3
=
1
3
∙
3
=
3
3
32
3
=
3
2 2
5
However, binomial denominator sometimes exists and
𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒 is being used to rationalize these denominators.
𝑎 + 𝑏 is the conjugate of 𝑎 − 𝑏 (and vice versa)
𝒂 + 𝒃 𝒂 − 𝒃 = 𝒂2 − 𝒃
𝑎 + 𝑏 is the conjugate of 𝑎 − 𝑏 (and vice versa)
𝒂+ 𝒃
𝒂− 𝒃 =𝒂−𝒃
2
3+ 5
=
=
2
∙
3− 5
3+ 5 3− 5
2 3− 5
3+ 5 3− 5
2 3− 5
=
9−5
2 3− 5
=
4
3− 5
2
5
3+ 2
𝑥
𝑥−2
32𝑥 7
72𝑥 8
3𝑥
8𝑥
36𝑦 2
3
3
3
3+ 3
𝑥+ 5
THANK YOU!
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