1. Lesson 10-1 Warm-Up

2. “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 = 64 • 3 64 is a perfect square and factor of 192. = 64 • 3 Multiplication Property of Square Roots = 8 3 Simplify 64

3. “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

4. 243 = 81 • 3 81 is a perfect square and a factor of 243.
Simplifying Radicals
LESSON 10-1
Additional Examples
Simplify
243 = • is a perfect square and a factor of 243.
= • Use the Multiplication Property of Square Roots.
= Simplify

5. 28x7 = 4x6 • 7x 4x6 is a perfect square and a factor of 28x7.
Simplifying Radicals
LESSON 10-1
Additional Examples
Simplify x7.
28x7 = 4x6 • 7x x6 is a perfect square and a factor of 28x7.
= 4x6 • 7x Use the Multiplication Property of Square Roots.
= 2x3 7x Simplify 4x6.

6. Simplify each radical expression.
Simplifying Radicals
LESSON 10-1
Additional Examples
Simplify each radical expression.
a •
12 • = • 32 Use the Multiplication Property of
Square Roots.
= Simplify under the radical.
= • 6 64 is a perfect square and a factor of 384.
= • 6 Use the Multiplication Property of
Square Roots.
= Simplify

7. 7 5x • 3 8x = 21 40x2 Multiply the whole numbers and
Simplifying Radicals
LESSON 10-1
Additional Examples
(continued)
b x • x
7 5x • x = x2 Multiply the whole numbers and
use the Multiplication Property of
Square Roots.
= x2 • 10 4x2 is a perfect square and a
factor of 40x2.
= x2 • Use the Multiplication Property of
Square Roots.
= 21 • 2x Simplify 4x2.
= 42x Simplify.

8. Suppose you are looking out a fourth floor window 52 ft
Simplifying Radicals
LESSON 10-1
Additional Examples
Suppose you are looking out a fourth floor window 52 ft
above the ground. Use the formula d = h to estimate the distance you can see to the horizon. Round your answer to the nearest mile.
d = h
= • 52 Substitute 52 for h.
= Multiply.
Use a calculator.
To the nearest mile, the distance you can see is 9 miles.

9. “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: a b a b

10. Simplify each radical expression.
Simplifying Radicals
LESSON 10-1
Additional Examples
Simplify each radical expression. a. 13 64 b. 49 x4
= Use the Division Property of Square Roots. 13 64
= Simplify 13 8
= Use the Division Property of Square Roots. 49 x4 7 x2
= Simplify and x4.

11. “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:

12. Simplify each radical expression.
Simplifying Radicals
LESSON 10-1
Additional Examples
Simplify each radical expression. a.
120 10
120 10
= Divide.
= 4 • 3 4 is a perfect square and a factor of 12.
= 4 • 3 Use the Multiplication Property of Square Roots.
= Simplify

13. = Divide the numerator and denominator by 3x.
Simplifying Radicals
LESSON 10-1
Additional Examples
(continued) b.
75x5
48x
= Divide the numerator and denominator by 3x.
75x5
48x
25x4 16
= Use the Division Property of Square Roots.
25x4 16
= Use the Multiplication Property of
Square Roots.
25 • x4 16
5x2 4
= Simplify , x4, and

14. “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:

15. Simplify by rationalizing the denominator.
Simplifying Radicals
LESSON 10-1
Additional Examples
Simplify by rationalizing the denominator. a. 3 7 3 7
√ 7
= • Multiply by to make the denominator a
perfect square.
3 7 49
= Use the Multiplication Property of Square Roots.
3 7 7
= Simplify

16. = • Multiply by to make the denominator a perfect square.
Simplifying Radicals
LESSON 10-1
Additional Examples
(continued) b. 11
12x3 √ 3x 11
12x3
= • Multiply by to make the denominator a
perfect square.
33x
36x4
= Use the Multiplication Property of Square Roots.
33x
6x2
= Simplify x4.

17. Simplify each radical expression.
Simplifying Radicals
LESSON 10-1
Lesson Quiz
Simplify each radical expression. • 12 36 3
8 2 48 2 a5
2 a a3 3x
15x3 5 5x