algebra 1 lesson 10-1 warm-up. algebra 1 “simplifying radicals” (10-1) what is a “radical...
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ALGEBRA 1
Lesson 10-1 Warm-Up
ALGEBRA 1
“Simplifying Radicals” (10-1)
What is a “radical expression”?
How can you simplify a radical expression?
Radical Expression: an expression that involves a radical (the expression inside or under the radical sign is called the radicand)
Example: 2 3 x + 3
To simplify a radical expression, remove perfect square factors from the radicand.
Rule: Multiplication Property of Square Roots: For every number a ≥ 0 and b ≥ 0:
ab = a • b
Example: 54 = 9 • 6 = 3 • 6 = 3 6
You can simplify radical expressions by rewriting them as a product of perfect square factors and the remaining factors
Example: Simplify 192 .
192 = 64 • 3 64 is a perfect square and factor of 192.
= 64 • 3 Multiplication Property of Square Roots
= 8 3 Simplify 64
ALGEBRA 1
“Simplifying Radicals” (10-1)
How can you simplify a radical expression that contains a variable?
You can simplify radical expressions containing variables with exponents of 2 or greater. A variable with an even exponent is a perfect square (Examples: n2 = n; n4 = n2;; n6 = n3). Therefore, you can use the Multiplication Property of Square Roots to simplify radical expressions containing variables as well (i.e. perfect square factors of the variable times the rest of the factors).
Example: Simplify 45a5
45a5 = 9 • a4 • 5 • a 9 and a4 are perfect square factors of 45a5.
= 9 • a4 • 5 • a Multiplication Property of Square Roots
= 3 • a2 • 5 • a Simplify 9a4
= 3a2 5a
ALGEBRA 1
243 = 81 • 3 81 is a perfect square and a factor of 243.
= 81 • 3 Use the Multiplication Property of Square Roots.
= 9 3 Simplify 81.
Simplifying RadicalsLESSON 10-1
Additional Examples
Simplify 243.
ALGEBRA 1
28x7 = 4x6 • 7x 4x6 is a perfect square and a factor of 28x7.
= 4x6 • 7x Use the Multiplication Property of Square Roots.
= 2x3 7x Simplify 4x6.
Simplifying RadicalsLESSON 10-1
Additional Examples
Simplify 28x7.
ALGEBRA 1
Simplify each radical expression.
a. 12 • 32
12 • 32 = 12 • 32 Use the Multiplication Property of
Square Roots.
= 384 Simplify under the radical.
= 64 • 6 64 is a perfect square and a factor of 384.
= 64 • 6 Use the Multiplication Property of
Square Roots.
= 8 6 Simplify 64.
Simplifying RadicalsLESSON 10-1
Additional Examples
ALGEBRA 1
(continued)
b. 7 5x • 3 8x
7 5x • 3 8x = 21 40x2 Multiply the whole numbers and
use the Multiplication Property of
Square Roots.
= 21 4x2 • 10 4x2 is a perfect square and a
factor of 40x2.
= 21 4x2 • 10 Use the Multiplication Property of
Square Roots.
= 42x 10 Simplify.
= 21 • 2x 10 Simplify 4x2.
Simplifying RadicalsLESSON 10-1
Additional Examples
ALGEBRA 1
To the nearest mile, the distance you can see is 9 miles.
8.83176 Use a calculator.
d = 1.5h
= 78 Multiply.
= 1.5 • 52 Substitute 52 for h.
Simplifying RadicalsLESSON 10-1
Additional Examples
Suppose you are looking out a fourth floor window 52 ft
above the ground. Use the formula d = 1.5h to estimate the
distance you can see to the horizon. Round your answer to the
nearest mile.
ALGEBRA 1
“Simplifying Radicals” (10-1)
What is the Division Property of Square Roots?
Rule: Division Property of Square Roots: This says that you van simplify the radical expressions of the numerator and denominator separately.
For every number a ≥ 0 and b 0:
=
Example:
Example:
Example:
ab
ab
ALGEBRA 1
Simplify each radical expression.
a. 1364
b. 49x4
= Use the Division Property of Square Roots.49x4
49
x4
= Use the Division Property of Square Roots.1364
13
64
= Simplify 64. 13
8
7
x2 = Simplify 49 and x4.
Simplifying RadicalsLESSON 10-1
Additional Examples
ALGEBRA 1
“Simplifying Radicals” (10-1)
Tip: When the denominator of a radicand is not a perfect square, it may be easier to divide the numerator by the denominator before simplifying the radicand.
Example:
Example:
ALGEBRA 1
Simplify each radical expression.
a. 120 10
= 4 • 3 4 is a perfect square and a factor of 12.
= 4 • 3 Use the Multiplication Property of Square Roots.
= 2 3 Simplify 4.
Simplifying RadicalsLESSON 10-1
Additional Examples
120 10
= 12 Divide.
ALGEBRA 1
b. 75x5
48x
= Divide the numerator and denominator by 3x.75x5
48x25x4
16
(continued)
= Use the Multiplication Property ofSquare Roots.
25 • x4
16
= Use the Division Property of Square Roots.25x4
16
5x2
4= Simplify 25, x4, and 16.
Simplifying RadicalsLESSON 10-1
Additional Examples
ALGEBRA 1
“Simplifying Radicals” (10-1)
What does it mean to “rationalize” the denominator?
Rationalize: If the denominator of a radical expression is not a perfect square, it is an irrational number (the square root of any number that is not a perfect square is irrational). To “rationalize” the denominator (make it into a perfect square), multiply both the numerator and denominator by the denominator to create an equal fraction in which the denominator is no longer in radical form.
Example:
Example:
ALGEBRA 1
Simplify by rationalizing the denominator.
a. 3 7
3 7 7
= Simplify 49.
3 7
49= Use the Multiplication Property of Square Roots.
Simplifying RadicalsLESSON 10-1
Additional Examples
3
7
3
7 7
7
√ 7
√ 7= • Multiply by to make the denominator a
perfect square.
ALGEBRA 1
(continued)
b. 11
12x3
√√ 3x
3x 3x
3x
11
12x3
11
12x3= • Multiply by to make the denominator a
perfect square.
33x
36x4= Use the Multiplication Property of Square Roots.
33x
6x2= Simplify 36x4.
Simplifying RadicalsLESSON 10-1
Additional Examples
ALGEBRA 1
12
36
Simplify each radical expression.
1. 16 • 8 2. 4 144 3.
4. 5. 2
a5
3x
15x3
8 2 48 3
3
2 a a3
5 5x
Simplifying RadicalsLESSON 10-1
Lesson Quiz