11-2 Radical Expressions
Preview
Warm Up
California Standards
Lesson Presentation
11-2 Radical Expressions
Warm Up
Identify the perfect square in each set.
1. 45 81 27 111 81 2. 156 99 8 25 25
3. 256 84 12 1000
256
4. 35 216 196 72 196
Write each number as a product of prime
numbers.
5. 36
6. 64
7. 196
8. 24
11-2 Radical Expressions
California
Standards
Extension of
2.0
Students understand and use such operations
as taking the opposite, finding the reciprocal,
taking a root, and raising to a fractional power.
They understand and use the rules of
exponents.
11-2 Radical Expressions
Vocabulary
radical expression
radicand
11-2 Radical Expressions
An expression that contains a radical sign
is a radical expression. There are many types
of radical expressions (such as square roots,
cube roots, fourth roots, and so on), but in this
chapter, you will study radical expressions that
contain only square roots.
Examples of radical expressions:
The expression under a radical sign is the
radicand. A radicand may contain numbers,
variables, or both. It may contain one term or
more than one term.
11-2 Radical Expressions
11-2 Radical Expressions
Remember that,
indicates a nonnegative square
root. When you simplify a square-root expression
containing variables, you must be sure your
answer is not negative. For example, you might
think that
but this is incorrect because
you do not know if x is positive or negative.
If x = 3, then
If x = –3, then
In both cases
simplification of
In this case,
In this case,
This is the correct
11-2 Radical Expressions
Additional Example 1: Simplifying Square-Root
Expressions
Simplify each expression.
A.
B.
C.
11-2 Radical Expressions
Check It Out! Example 1
Simplify each expression.
a.
b.
11-2 Radical Expressions
Check It Out! Example 1
Simplify each expression.
c.
d.
11-2 Radical Expressions
11-2 Radical Expressions
Additional Example 2A: Using the Product
Property of Square Roots
Simplify. All variables represent nonnegative
numbers.
Factor the radicand using perfect
squares.
Product Property of Square Roots
Simplify.
11-2 Radical Expressions
Additional Example 2B: Using the Product Property
of Square Roots
Simplify. All variables represent nonnegative
numbers.
Product Property of Square Roots
Product Property of Square Roots
Since x is nonnegative,
.
11-2 Radical Expressions
Helpful Hint
When factoring the radicand, use factors that are
perfect squares. In Example 2A, you could have
factored 18 as 6  3, but this contains no perfect
squares.
11-2 Radical Expressions
Check It Out! Example 2a
Simplify. All variables represent nonnegative
numbers.
Factor the radicand using perfect
squares.
Product Property of Square Roots
Simplify.
11-2 Radical Expressions
Check It Out! Example 2b
Simplify. All variables represent nonnegative
numbers.
Product Property of Square Roots
Product Property of Square Roots
Since y is nonnegative,
.
11-2 Radical Expressions
Check It Out! Example 2c
Simplify. All variables represent nonnegative
numbers.
Factor the radicand using perfect
squares.
Product Property of Square Roots
Simplify.
11-2 Radical Expressions
11-2 Radical Expressions
Additional Example 3: Using the Quotient
Property of Square Roots
Simplify. All variables represent nonnegative
numbers.
B.
A.
Quotient Property
of Square
Roots
Simplify.
Simplify.
Quotient
Property of
Square Roots
Simplify.
11-2 Radical Expressions
Check It Out! Example 3
Simplify. All variables represent nonnegative
numbers.
a.
b.
Simplify.
Quotient Property
of Square
Roots
Simplify.
Quotient
Property of
Square
Roots
Simplify.
11-2 Radical Expressions
Check It Out! Example 3c
Simplify. All variables represent nonnegative
numbers.
Quotient Property of
Square Roots
Factor the radicand using
perfect squares.
Simplify.
11-2 Radical Expressions
Additional Example 4A: Using the Product and
Quotient Properties Together
Simplify. All variables represent nonnegative
numbers.
Quotient Property
Product Property
Write 108 as 36(3).
Simplify.
11-2 Radical Expressions
Additional Example 4B: Using the Product and
Quotient Properties Together
Simplify. All variables represent nonnegative
numbers.
Quotient Property
Product Property
Simplify.
11-2 Radical Expressions
Caution!
In the expression
common factors.
simplified.
and 5 are not
is completely
11-2 Radical Expressions
Check It Out! Example 4a
Simplify. All variables represent nonnegative
numbers.
Quotient Property
Product Property
Write 20 as 4(5).
Simplify.
11-2 Radical Expressions
Check It Out! Example 4b
Simplify. All variables represent nonnegative
numbers.
Quotient Property
Product Property
Write
Simplify.
as
.
11-2 Radical Expressions
Check It Out! Example 4c
Simplify. All variables represent nonnegative
numbers.
Quotient Property
Simplify.
11-2 Radical Expressions
Additional Example 5: Application
Quadrangle
250
A quadrangle on a college campus
is a square with sides of 250 feet.
If a student takes a shortcut by
walking diagonally across the
quadrangle, how far does he walk?
Give the answer as a radical
expression in simplest form. Then
estimate the length to the nearest
tenth of a foot.
250
The distance from one corner of the square to the
opposite one is the hypotenuse of a right triangle.
Use the Pythagorean Theorem: c2 = a2 + b2.
11-2 Radical Expressions
Additional Example 5 Continued
Solve for c.
Substitute 250 for a and b.
Simplify.
Factor 125,000 using
perfect squares.
11-2 Radical Expressions
Additional Example 5 Continued
Use the Product Property of
Square Roots.
Simplify.
Use a calculator and round
to the nearest tenth.
The distance is
ft, or about 353.6 feet.
11-2 Radical Expressions
Check It Out! Example 5
A softball diamond is a square
with sides of 60 feet. How long
is a throw from third base to
first base in softball? Give the
answer as a radical expression
in simplest form. Then estimate
the length to the nearest tenth
of a foot.
The distance from one corner of the square to the
opposite one is the hypotenuse of a right triangle.
Use the Pythagorean Theorem: c2 = a2 + b2.
11-2 Radical Expressions
Check It Out! Example 5 Continued
Solve for c.
Substitute 60 for a and b.
Simplify.
Factor 7,200 using
perfect squares.
11-2 Radical Expressions
Check It Out! Example 5 Continued
Use the Product Property
of Square Roots.
Simplify.
Use a calculator and round
to the nearest tenth.
The distance is
, or about 84.9 feet.
11-2 Radical Expressions
Lesson Quiz: Part I
Simplify each expression.
1.
2.
6
|x + 5|
Simplify. All variables represent nonnegative
numbers.
3.
4.
5.
6.
11-2 Radical Expressions
Lesson Quiz: Part II
7. Two archaeologists leave from
the same campsite. One
travels 10 miles due north
and the other travels 6 miles
due west. How far apart are
the archaeologists? Give the
answer as a radical
expression in simplest form.
Then estimate the distance to
the nearest tenth of a mile.
mi; 11.7 mi