Simplifying Radical Expressions Mathemat

RADICALS
Presented by:
Josaiah Mae Gonzaga
Kerztein Acompañado
9th grade

3
81
Radical Symbol
Radicand
Radical
Index
If there are no
index, it is
assumed to be 2.
Parts of a Radical

Laws of Radicals
Laws of radicals are similar to the laws of integral exponents.
Let a, b, m, n are integers:

Laws Proof
1. The nth root of a product is the product of the nth
roots
𝑛
𝑥𝑦 =𝑛
𝑥 𝑛
𝑦
𝑛
𝑥𝑦 = 𝑥𝑦
1
𝑛 If x is a real number and m is a positive
integer greater than 1, then 𝑚
𝑥 = 𝑥
1
𝑚 = 𝑥
1
𝑛 𝑦
1
𝑛 Power
Rule for Products = 𝑛
𝑥 𝑛
𝑦
2.The nth root of a quotient is the quotient of the nth
roots.
𝑛 𝑥
𝑦
=
𝑛
𝑥
𝑛 𝑦
𝑛 𝑥
𝑦
=
𝑥
𝑦
1
𝑛
If x is a real number and m is a positive
integer greater than 1, then 𝑚
𝑥 = 𝑥
1
𝑚 =
𝑥
1
𝑛
𝑦
1
𝑛
Power Rule
for Quotients =
𝑛
𝑥
𝑛 𝑦
3.
𝑚 𝑛
𝑥 =𝑚𝑛
𝑥 𝑚 𝑛
𝑥 =
𝑚
𝑥
1
𝑛= 𝑥
1
𝑛
1
𝑚 = 𝑥
1
𝑛
1
𝑚 Power of a power
property
= 𝑥
1
𝑛𝑚= 𝑚𝑛
𝑥
4. If x is a real number and n is odd, then 𝑛
𝑥 = 𝑥
5. If x is a real number and n is even, then
𝑛
𝑥𝑛 = 𝑥

Examples:
Apply the laws of radicals:
1. (
3
12)3 = 12
1
3 13
= 12
1
3 = 12
2 . (
5
4) . (5 8) =
5
(4)(8) =
5
32 = 12 , multiply if the indices are the same, just
multiply the radicand, copy the
common index, then simplify.
3.
18
2 =
18
2
divide the radicands if the indices are the same, then simplify
by extracting the square root of 9 The 9 = 3, since 32 = 9
The fifth root of 32 is 2, meaning, when you multiply the root 2 by itself 5 times the answer is 32.
The index 5 indicates the number of times the root be multiplied by itself the radicand. In this to
get case, 25 = 32 Непсе
5
32 = 2
= 9 = 3

Examples:
Apply the laws of radicals:
4.
3
64 = = 2 , the square root of 64 is 8 because the 82 = 64 and
the cube root of 8 = 2, since : 23 = 8.
3
8
5.
8
2 =
8
2
= 4 = 2
divide the radicands if the indices are the same, then simplify
by extracting the square root of 4. The 4 = 2, since 22 = 4

Simplifying
Radical Expressions

Objective:
• Simplify radical expression using the laws of
radicals.

Here are some examples of numbers that are not perfect square:
8 = 4 . 2 = 2 2
12 = 4 . 3 = 2 3
18 = 9 . 2 = 3 2
32 = 16 . 2 = 4 2
40 = 4 . 10 = 2 10
44 = 4 . 11 = 2 11
20 = 4 . 5 = 2 5
24 = 4 . 6 = 2 6
27 = 9 . 3 = 3 3
28 = 4 . 7 = 2 7
45 = 9 . 5 = 3 5
48 = 16 . 3 = 4 3
50 = 25 . 2 = 5 2
63 = 9 . 7 = 3 7
68 = 4 . 17 = 2 17
72 = 36 . 2 = 6 2
52 = 4 . 13 = 2 13
54 = 9 . 6 = 3 6
56 = 4 . 14 = 2 14
60 = 4 . 15 = 2 15
75 = 25 . 3 = 5 3
76 = 4 . 19 = 2 19
80 = 16 . 5 = 4 7
96 = 16 . 6 = 4 6
98 = 49 . 2 = 7 2
99 = 9 . 11 = 3 11
84 = 4 . 21 = 2 21
88 = 4 . 22 = 2 22
90 = 9 . 10 = 3 10
92 = 4 . 23 = 2 23

1.
𝑛
𝑎𝑏 =
𝑛
𝑎𝑏 .
𝑛
𝑎𝑏, the nth root of a product is equal to the nth root of each
factor.
2. 𝑛 𝑚
𝑝 = 𝑚𝑛
𝑝, the nth root of the mth root of a number is equal to the nmth
root of the number.
3.
𝑛 𝑝
𝑞
=
𝑛 𝑝
𝑛 𝑞
, the nth root of the quotient is equal to the nth root of the
numerator and the nth root of the denominator.
The properties of radicals which can be useful in simplifying radical
expressions are as follows:

Simplifying Radical Expressions by:

Reducing the Radicand
01
• Reducing radicand is finding a factor of a radicand whose indicated
roots can be found.

Example:
1. Simplify 50
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
, the factors of 50 are 25 and 2 . 25 is a factor which a
perfect square, meaning you can extract the square root
, the square root of the product is equal to the square root
of the factors.
, the square root of 25 is 5, square root of 2 is already in
simplified form.
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 50 = 5 2 in simplest form
50 is not a perfect square, therefore you have to find a factor of
50 which is a perfect square.
50 = 25. 2
= 25 . 2
= 5 2

2. Simplify
3
81
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: The factor of 81 which is a perfect cube is 27.
3
81
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒
3
81 = 3
3
3 since 3
3
3
3
= 33 3
1
3
3
= 27(3) = 81
=
3
27 .
3
3 = 3
3
3
=
3
27. 3

Another example:
𝑛
𝑎𝑏 = 𝑛
𝑎 .
𝑛
𝑏
1. 8𝑥5𝑦6𝑧13 4.2. 𝑥4. 𝑥. 𝑦6. 𝑧12. 𝑧
=
=
2𝑥2𝑦3𝑧6 2. 𝑥. 𝑧
= 2𝑥2𝑦3𝑧6 2𝑥𝑧

Activity #1.
Simplify the following.
1. 63
2. 99
3.
3
24
4.
3
40
5. 75

1. 63
2. 99
3.
3
24
4.
3
40
5. 75
63 = 9 . 7
= 9 . 7
= 3 7
99 = 9 . 11
= 9 . 11
= 3 11
3
24=
3
8 . 3
=
3
23 .
3
3
= 23
3
=
3
23. 3
Answer key: (2 points each)
3
40=
3
8 . 5
=
3
23 .
3
5
= 23
5
=
3
23. 5
63 = 25 . 3
= 25 . 3
= 5 3

Reducing the order of the
Radicals
02
• To reduce the order of radicals is to reduce the index to its
lowest possible number

Example:
1. Simplify
8
𝑎12
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: , find a factor of radicand that is a power of 8
, find the 8th root of each factor
, reduce the rational exponent to lowest term
8
𝑎12 =
8
𝑎8. 𝑎4
=
8
𝑎8 .
8
𝑎4
= 𝑎
8
𝑎4
= 𝑎 . 𝑎
4
8
= 𝑎 𝑎
, change the radical to exponential form
, change the exponential form to radical form.
= 𝑎 . 𝑎
1
2

Reducing the index to the lowest possible order
𝑚
𝑛
𝑎 = 𝑚𝑛
𝑎 .
𝑛 𝑚
𝑎
Example:
1.
20
32𝑚15𝑛5 =
= 4 5
25 𝑚3 5𝑛5
= 4
2𝑚3𝑛
4 5
25 𝑚3 5𝑛5

Simplifying each radical
Example:
1. 𝑥5 =
=
𝑥4 . 𝑥
=
𝑥2 𝑥
𝑥4 . 𝑥 2.
3
𝑏7 =
3
𝑏6 . 𝑏
3
𝑏6 .
3
𝑏
𝑏2 3
𝑏
=
=

A. Simplify the following.
1. 5
𝑥6 = 5
𝑥5 . 𝑥 = x5
𝑥
2. 36𝑎2 = 36 𝑎2
3.
3
80𝑎10𝑏16 =
3
8. 10𝑎9. 𝑎 . 𝑏15. 𝑏 =
3
23 . 10𝑎9 . 𝑎. 𝑏15. 𝑏
= 62 a = 6𝑎
=
3
23 3
𝑎9 3
𝑏153
10𝑎𝑏 = 2𝑎3𝑏53
10𝑎𝑏

Rationalizing the
Denominator
03

Objective:
• Rationalize a fraction whose denominator
contains a radical.

Rationalizing the denominator of the Radicand
• To rationalize the denominator, we multiply the
denominator by an appropriate expression such
that the product will be a perfect nth root.
• When the denominator is a binomial, we multiply it
by its conjugate. The conjugate of a binomial is a
binomial of the same terms but of different sign.

Simplify:
1)
1
36
2)
3
64 5)
5
4
3)
4
9
4)
1
49
No need to rationalize
because the radical sign is
on the numerator.
=
1
6
=
3
8
=
4
9
=
2
3
=
1
49
=
1
7
=
5
4
=
5
2

Simplify by rationalizing the denominator of the Radicand :
1)
1
7
=
1
7
.
7
7
=
7
7 2
2)
2
𝑥
=
2
𝑥
.
𝑥
𝑥
=
2 𝑥
𝑥 2
3)
2𝑥
𝑦
=
2𝑥
𝑦
.
𝑦
𝑦
=
2𝑥𝑦
𝑦 2
=
7
7
=
2 𝑥
𝑥
=
2𝑥𝑦
𝑦

Simplify by rationalizing the denominators.
1)
5
7
2)
8𝑥
3𝑦
3)
3𝑚
2𝑛
=
5 7
7
=
2 6𝑥𝑦
3𝑦
=
6𝑚𝑛
2𝑛
=
5
7
.
7
7
=
5 7
7 2
=
8𝑥
3𝑦
.
3𝑦
3𝑦
=
24 𝑥𝑦
3𝑦 2 =
4 . 6𝑥𝑦
3𝑦 2
=
3𝑚
2𝑛
.
2𝑛
2𝑛
=
6𝑚𝑛
2𝑛 2
Activity #1

Conjugate of Radical Expressions
Examples:
Radical Expression Conjugate
3 + 2
5 - 2
3 - 2
5 + 2
The product of the radical expression and its conjugate is an integer. In finding its
product is the same procedure in multiplying the sum and difference of 2
binomials. Like, (x + y)(x - y) = x2 - y2

Addition and Subtraction
of Radicals
04

Objective:
• Differentiate like radicals from unlike radicals;
• Add and subtract radical expressions.

Like Radicals
Unlike Radicals
2 3 and -5 3
3 𝑥 and -2 𝑥
5x 2𝑦 and 3y 2𝑦
83
𝑥𝑦 and 3
𝑥𝑦
Same index and same radicand
2 and 6
3
7𝑥 and 7𝑥
4 2 and 4 3 8 and
3
8
Same index and different radicands
Different index and same radicands

Simplify.
1. 3 2 - 5 2
2. 3 𝑎 - 𝑎 + 3 𝑎
3. 6 3𝑎 - 2 𝑎 + 3𝑎 + 5 𝑎
= (3 - 5) 2 = -2 2
= (2 – 1 + 3) 𝑎 = 4 𝑎
= (6 + 1) 3𝑎 + (-2 + 5) 𝑎
= 7 3𝑎 + 3 𝑎

4. 8 + 8 8 + 18 = (1 + 1 + 1) 8 + 18
= 9 2
= 3 8 + 18
= 3 4 . 2 + 9 . 2
= 3 4 2 + 9 2
= 3 . 2 2 +3 2
= 6 2 +3 2

5. 8 + 18 −32 = 4.2 + 9.2 - 16.2
= 2
= 2 2 + 3 2 - 4 2
= (2 + 3 - 4) 2
6. 12𝑎3 - 300𝑎3 = 4𝑎2. 3𝑎 - 100𝑎2. 3𝑎
= 4𝑎2 3𝑎 - 100𝑎2 3𝑎
= 2𝑎 3𝑎 - 10𝑎 3𝑎
= −8a 3𝑎

Find the sum and difference of the following radical expression by matching
column B with column A.
Activity #3: Add, Subtract then Match

Group activity:
Group the following radicals with same radicand, with different
radicand and place them inside each box.
1. 11 3 + 5 3
2. 7 8 + 3 18 - 2 2
3. 5 12 + 5 27 - 4 3
4. 8
2𝑦
7
+ 3
2𝑦
7
-
2𝑦
7
5. 9
3𝑦
25
+
32𝑦
5
- 3
3𝑦
5

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