Simplifying Radical Expressions Simplifying Radicals Radicals with variables.
15.1 – Introduction to Radicals
17
15.1 – Introduction to Radicals Radical Expressions Finding a root of a number is the inverse operation of raising a number to a power. This symbol is the radical or the radical s n a inde x radical sign radica nd The expression under the radical sign is the radicand. The index defines the root to be taken.
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15.1 – Introduction to Radicals. Radical Expressions. Finding a root of a number is the inverse operation of raising a number to a power. radical sign. index. radicand. This symbol is the radical or the radical sign. The expression under the radical sign is the radicand. - PowerPoint PPT Presentation
Transcript of 15.1 – Introduction to Radicals
15.1 – Introduction to RadicalsRadical Expressions
Finding a root of a number is the inverse operation of raising a number to a power.
This symbol is the radical or the radical sign
n aindex
radical sign
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
Square Roots
If a is a positive number, then
a is the positive square root of a and
100
a is the negative square root of a.
A square root of any positive number has two roots – one is positive and the other is negative.
Examples:
10
25
49
5
7
11 36 6
9 non-real #
15.1 – Introduction to Radicals
81.0 9.0
What does the following symbol represent?
The symbol represents the positive or principal root of a number.
15.1 – Introduction to Radicals
4 5xyWhat is the radicand of the expression ?
5xy
What does the following symbol represent?
The symbol represents the negative root of a number.
15.1 – Introduction to Radicals
3 525 yxWhat is the index of the expression ?
3
Cube Roots
3 27
A cube root of any positive number is positive.
Examples:
3 5
43
125
64
3 8 2
A cube root of any negative number is negative.
3 a
15.1 – Introduction to Radicals
3 27 3 3 8 2
nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
2
4 81 3
4 16
5 32 2
43 81
42 16
52 32
15.1 – Introduction to Radicals
nth Roots
4 16
An nth root of any number a is a number whose nth power is a.
Examples:
15 1
Non-real number
6 1 Non-real number
3 27 3
15.1 – Introduction to Radicals
Radicals with Variables
8z
Examples:
123 8y
20x 64x
9 243 64x y 3 84x y
4z 10x 32x
42y
26x 12x 53y 15y 33 7x y 9 21x y
15.1 – Introduction to Radicals
12x 6x 5 15y 3y 3 219 yx 73yx
15.2 – Simplifying RadicalsSimplifying Radicals using the Product Rule
40
Examples:
18
700
4 10
9 2
100 7
If and are real numbers, then a ba b a b
Product Rule for Square Roots
2 10
3 2
10 7
15 157 75 7 25 3 7 5 3 35 3
104
29
Simplifying Radicals using the Quotient Rule
16
81
Examples:
2
5
4
9
aIf and are real numbers and 0, then
b
aa b b
b
Quotient Rule for Square Roots
2
25
9 5
7
3 5
7
16
81
2
25
45
49
15.2 – Simplifying Radicals
49
45
Simplifying Radicals Containing Variables
11x
Examples:
8
27
x
67
25
y y
3 7
5
y y
10x x 5x x
418x 49 2x 23 2x
8
9 3
x
4
3 3
x8
27
x
15.2 – Simplifying Radicals
25
7 7y
25
7 7y
Simplifying Cube Roots
3 88
Examples:
381
8
310
27
3
3
81
8
3 27 3
2
3 8 11 32 11
3 50 3 50
3 10
3
3
3
10
27
33 3
2
15.2 – Simplifying Radicals
3 3 727m n
Examples:
3 3 63 m n n
2 33mn n
15.2 – Simplifying Radicals
Examples:
12 4 185 64x y z
10 2 4 15 35 32 2x x y z z
2 3 2 4 352 2x z x y z
15.2 – Simplifying Radicals
5 3x x
Review and Examples:
6 11 9 11
8x
15 11
12 7y y 5y
7 3 7 2 7
15.3 – Adding and Subtracting Radicals
27 75
Simplifying Radicals Prior to Adding or Subtracting
3 20 7 45
9 3 25 3
3 4 5 7 9 5
3 3 5 3 8 3
3 2 5 7 3 5
6 5 21 5 15 5
36 48 4 3 9 6 16 3 4 3 3
6 4 3 4 3 3 3 8 3
15.3 – Adding and Subtracting Radicals
4 3 39 36x x x
Simplifying Radicals Prior to Adding or Subtracting
6 63 310 81 24p p
2 2 23 6x x x x x
23 6x x x x x 23 5x x x
6 63 310 27 3 8 3p p
2 23 310 3 3 2 3p p 2 328 3p
2 23 330 3 2 3p p
15.3 – Adding and Subtracting Radicals
