Download Multiplying Square Roots

CHAPTER 5
Number Theory and the
Real Number System
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5.4
The Irrational Numbers
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1.
2.
3.
4.
Objectives
Define the irrational numbers.
Simplify square roots.
Perform operations with square roots.
Rationalize the denominator.
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The Irrational Numbers
• The set of irrational numbers is the set of numbers
whose decimal representations are neither terminating
nor repeating.
For example, a well-known irrational number is π
because there is no last digit in its decimal
representation, and it is not a repeating decimal:
π ≈ 3.1415926535897932384626433832795…
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Square Roots
• The principal square root of a nonnegative number n,
written n , is the positive number that when
multiplied by itself gives n.
For example,
36  6 because 6 · 6 = 36.
Notice that 36 is a rational number because 6 is a
terminating decimal.
Not all square roots are irrational.
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Square Roots
• A perfect square is a number that is the square of a
whole number.
For example, here are a few perfect squares:
0 = 02
1 = 12
4 = 22
9 = 32
The square root of a perfect square is a whole number:
0  0, 1  1, 4  2, 9  3
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The Product Rule For Square Roots
If a and b represent nonnegative numbers, then
ab  a  b and a  b  ab .
The square root of a product is the product of the square
roots.
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Example 1: Simplifying Square Roots
Simplify, if possible:
a. 75
75  25  3
c. 17
b.
500
500  100  5
 25  3
 100  5
5 3
 10 5
Because 17 has no perfect square
factors (other than 1), it cannot be
simplified.
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Multiplying Square Roots
If a and b are nonnegative, then we can use the product
rule
a  b  a b
to multiply square roots.
The product of the square roots is the square root of the
product.
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Example 2: Multiplying Square Roots
Multiply:
a. 2  5  2  5  10
b.
7  7  49  7
c.
6  12  6 12  72  36  2  36  2  6 2
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Dividing Square Roots
The Quotient Rule
If a and b represent nonnegative real numbers and b ≠ 0,
then
a
a

b
b
and
a
a

.
b
b
The quotient of two square roots is the square root of
the quotient.
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Example 3: Dividing Square Roots
Find the quotient:
a. 75
75
3
b.

3
 25  5
90
90

 45  9  5  9  5  3 5
2
2
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Adding and Subtracting Square Roots
• The number that multiplies a square root is called the
square root’s coefficient.
• Square roots with the same radicand can be added or
subtracted by adding or subtracting their coefficients:
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Example 4: Adding and Subtracting Square
Roots
Add or subtract as indicated:
a. 7 2  5 2
b. 2 5  6 5
Solution:
a. 7 2  5 2  (7  5) 2  12 2
b. 2 5  6 5  (2  6) 5  4 5
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Rationalizing the Denominator
• We rationalize the denominator to rewrite the
expression so that the denominator no longer contains
any radicals.
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Example 6: Rationalizing Denominators
Rationalize the denominator:
a. 15 15 6 15 6 5 6
6
b.

6

6

6

2
3
3
3 5
15
15





5
5
5
5 5
25
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