Simplifying a Radical Review Simplify each radical and leave the answer in exact form. 1. 2. 3.
10.3 Simplifying Radical Expressions
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Transcript of 10.3 Simplifying Radical Expressions
10.3 Simplifying Radical Expressions
Use the product rule for radicals.
Objective 1
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Product Rule for Radicals
If and are real numbers and n is a natural number, then
That is, the product of two nth roots is the nth root of the product.
n a n b.n n na b ab
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Use the product rule for radicals.
Use the product rule only when the radicals have the same index.
Multiply. Assume that all variables represent positive real numbers.
5 13 5 13
7 xy 7xy
65
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CLASSROOM EXAMPLE 1 Using the Product Rule
Solution:
Multiply. Assume that all variables represent positive real numbers.
3 32 7 3 2 7
66 2 38 2r r 6 516r
2 25 59 8y x xy 4 25 72y x
37 5
3 14
This expression cannot be simplified by using the product rule.
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CLASSROOM EXAMPLE 2 Using the Product Rule
Solution:
Use the quotient rule for radicals.
Objective 2
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Quotient Rule for Radicals
If and are real numbers, b ≠ 0, and n is a natural number,
then
That is, the nth root of a quotient is the quotient of the nth roots.
n a n b
.n
nn
a ab b
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Use the quotient rule for radicals.
Simplify. Assume that all variables represent positive real numbers.
10081
109
1125
115
8
16y
318125
3
3
18125
23
12
xr
3 2
3 12
x
r
3 185
8
16y
4
4y
3 2
4
xr
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CLASSROOM EXAMPLE 3 Using the Quotient Rule
Solution:
Simplify radicals.
Objective 3
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Conditions for a Simplified Radical
1. The radicand has no factor raised to a power greater than or equal to the index.
2. The radicand has no fractions.
3. No denominator contains a radical.
4. Exponents in the radicand and the index of the radical have greatest common factor 1.
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Simplify radicals.
Be careful with which factors belong outside the radical sign and which belong inside.
Simplify.
32 216 216 4 2
300 3100 3100 10 3
35 Cannot be simplified further.
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CLASSROOM EXAMPLE 4 Simplifying Roots of Numbers
Solution:
3 54 3 227 33 27 2 33 2
4 243 44 3 3 43 3
Simplify. Assume that all variables represent positive real numbers.
725p 3 225 ( )p p 35p p
372y x 2236 y y x 6 2y yx
7 5 63 27y x z3 6 3 2 63 3 y y x x z 2 2 233y xz yx
5 74 32a b4 4 4 34 2 2 a a b b 342 2ab ab
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CLASSROOM EXAMPLE 5 Simplifying Radicals Involving Variables
Solution:
Simplify. Assume that all variables represent positive real numbers.
12 32 1/1232 1/ 42
6 2t 1/62t 3 t
4 2
1/3t
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CLASSROOM EXAMPLE 6 Simplifying Radicals by Using Smaller Indexes
Solution:
If m is an integer, n and k are natural numbers, and all indicated roots exist, then
.kn nkm ma a
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Simplify radicals.
kn kma
Simplify products and quotients of radicals with different indexes.
Objective 4
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Simplify.
35 4
1/ 2 3/ 6 6 3 6 125 5 5 5
1/3 2/ 6 6 2 6 164 4 4
35 4 6 6125 16 6 2000
The indexes, 2 and 3, have a least common index of 6, use rational exponents to write each radical as a sixth root.
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CLASSROOM EXAMPLE 7 Multiplying Radicals with Different Indexes
Solution:
Use the Pythagorean theorem.
Objective 5
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Pythagorean TheoremIf c is the length of the longest side of a right triangle and a and b are lengths of the shorter sides, then
c2 = a2 + b2.
The longest side is the hypotenuse, and the two shorter sides are the legs, of the triangle. The hypotenuse is the side opposite the right angle.
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Use the Pythagorean theorem.
Find the length of the unknown side of the triangle.
c2 = a2 + b2
c2 = 142 + 82
c2 = 196 + 64c2 = 260
260c
4 65c
4 65c
2 65c The length of the hypotenuse is 2 65.
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CLASSROOM EXAMPLE 8 Using the Pythagorean Theorem
Solution:
Find the length of the unknown side of the triangle.
c2 = a2 + b2
62 = 42 + b2
36 = 16 + b2
20 = b2
20 b
4 5 b
4 5 b
2 5 bThe length of the leg is 2 5.
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CLASSROOM EXAMPLE 8 Using the Pythagorean Theorem (cont’d)
Solution:
Use the distance formula.
Objective 6
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Distance Formula
The distance between points (x1, y1) and (x2, y2) is
2 22 1 2 1( ) ( )d x x y y
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Use the distance formula.
Find the distance between each pair of points. (2, –1) and (5, 3)
Designate which points are (x1, y1) and (x2, y2).(x1, y1) = (2, –1) and (x2, y2) = (5, 3)
2 22 1 2 1( ) ( )d x x y y
2 2( ) (5 32 ( ))1d
2 2(3) (4)d
9 16d
5 52d
Start with the x-value and y-value of the same point.
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CLASSROOM EXAMPLE 9 Using the Distance Formula
Solution:
Find the distance between each pairs of points. (–3, 2) and (0, 4)
Designate which points are (x1, y1) and (x2, y2).(x1, y1) = (–3, 2) and (x2, y2) = (0, 4)
2 22 1 2 2( ) ( )d x x y y
2 2( ( )) (3 )0 4 2d
2 2(3) ( 6)d
9 36d
3 545d Slide 10.3- 24
CLASSROOM EXAMPLE 9 Using the Distance Formula (cont’d)
Solution: