1. 243 = 81 • 3 81 is a perfect square and a factor of 243.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
Simplify
243 = • is a perfect square and a factor of 243.
= • Use the Multiplication Property of Square Roots.
= Simplify
Quick Check
11-1

2. 28x7 = 4x6 • 7x 4x6 is a perfect square and a factor of 28x7.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
Simplify x7.
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.
Quick Check
11-1

3. Simplify each radical expression.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
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
11-1

4. 7 5x • 3 8x = 21 40x2 Multiply the whole numbers and
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
(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.
Quick Check
11-1

5. To the nearest mile, the distance you can see is 9 miles.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
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.
Quick Check
11-1

6. Simplify each radical expression.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
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.
Quick Check
11-1

7. Simplify each radical expression.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
Simplify each radical expression. a.
120 10
= Divide.
120 10
= 4 • 3 4 is a perfect square and a factor of 12.
= 4 • 3 Use the Multiplication Property of Square Roots.
= Simplify
11-1

8. = Divide the numerator and denominator by 3x.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
(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
= Simplify , x4, and
5x2 4
Quick Check
11-1

9. Simplify each radical expression.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
Simplify each radical expression. a. 3 7 3 7
= • Multiply by to make the denominator a
perfect square.
= Use the Multiplication Property of Square Roots.
3 7 49
= Simplify
3 7 7
11-1

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

11. Simplify each radical expression.
Simplifying Radicals
ALGEBRA 1 LESSON 11-1
Simplify each radical expression. • 12 36 3
8 2 48 2 a5
2 a a3 3x
15x3 5 5x
11-1

12. Operations with Radical Expressions
ALGEBRA 1 LESSON 11-2
Simplify
= Both terms contain
= (4 + 1) 3 Use the Distributive Property to
combine like radicals.
= Simplify.
Quick Check
11-2

13. Operations with Radical Expressions
ALGEBRA 1 LESSON 11-2
Simplify –
– = • 5 9 is a perfect square and a factor of 45.
= – 9 • 5 Use the Multiplication Property of
Square Roots.
= – Simplify
= (8 – 3) 5 Use the Distributive Property to
combine like terms.
= Simplify.
Quick Check
11-2

14. Operations with Radical Expressions
ALGEBRA 1 LESSON 11-2
Simplify 5( ).
5( ) = Use the Distributive Property.
= 4 • Use the Multiplication Property
of Square Roots.
= Simplify.
Quick Check
11-2

15. Operations with Radical Expressions
ALGEBRA 1 LESSON 11-2
Simplify ( 6 – )( ).
( 6 – )( )
= – – Use FOIL.
= 6 – – 3(21) Combine like radicals and
simplify and
= 6 – • 14 – 63 9 is a perfect square factor of 126.
= 6 – • – 63 Use the Multiplication Property of
Square Roots.
= 6 – – 63 Simplify
= –57 – Simplify.
Quick Check
11-2

16. Operations with Radical Expressions
ALGEBRA 1 LESSON 11-2
Simplify . 8
7 – 3
= • Multiply the numerator and
denominator by the conjugate
of the denominator. 8
7 – 3
= Multiply in the denominator.
8( )
7 – 3
= Simplify the denominator.
8( ) 4
= 2( ) Divide 8 and 4 by the common
factor 4.
= Simplify the expression.
Quick Check
11-2

17. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
Solve each equation. Check your answers.
a x – 5 = 4
x – 5 = 4
x = 9 Isolate the radical on the left side of the equation.
( x)2 = 92 Square each side.
x = 81
4 = 4
Check: x – 5 = 4
81 – Substitute 81 for x.
9 –
11-3

18. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
(continued)
b x – 5 = 4
( x – 5)2 = 42 Square each side.
x – 5 = 16 Solve for x.
x = 21
Check: x – 5 = 4
21– 5 = 4 Substitute 21 for x.
16 = 4
4 = 4
Quick Check
11-3

19. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
On a roller coaster ride, your speed in a loop depends on the height of the hill you have just come down and the radius of the loop in feet. The equation v = 8 h – 2r gives the velocity v in feet per second
of a car at the top of the loop.
11-3

20. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
(continued)
The loop on a roller coaster ride has a radius of 18 ft.
Your car has a velocity of 120 ft/s at the top of the loop.
How high is the hill of the loop you have just come
down before going into the loop?
Solve v = 8 h – 2r for h when v = 120 and r = 18.
120 = 8 h – 2(18) Substitute 120 for v and 18 for r.
= Divide each side by 8 to isolate the radical.
15 = h – 36 Simplify.
8 h – 2(18) 8
120
(15)2 = ( h – 36)2 Square both sides.
225 = h – 36
261 = h
Quick Check
The hill is 261 ft high.
11-3

21. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
Solve 3x – 4 = 2x + 3.
( 3x – 4)2 = ( 2x + 3)2 Square both sides.
3x – 4 = 2x + 3 Simplify.
3x = 2x + 7 Add 4 to each side.
x = 7 Subtract 2x from each side.
Check: x – 4 = 2x + 3
3(7) – (7) + 3 Substitute 7 for x.
17 = 17
The solution is 7.
Quick Check
11-3

22. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
Solve x = x + 12.
(x)2 = ( x + 12)2 Square both sides.
x2 = x + 12
x2 – x – 12 = 0 Simplify.
(x – 4)(x + 3) = 0 Solve the quadratic equation by factoring.
(x – 4) = 0 or (x + 3) = 0 Use the Zero–Product Property.
x = 4  or   x = –3 Solve for x.
Check: x = x + 12
– –3 + 12
4 = 4 –3 = 3 /
The solution to the original equation is 4.
The value –3 is an extraneous solution.
Quick Check
11-3

23. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
Solve 3x + 8 = 2.
3x = –6
( 3x)2 = (–6) Square both sides.
3x = 36
x = 12
Check: x + 8 = 2
3(12)    Substitute 12 for x.
= 2   x = 12 does not solve the original equation. /
3x + 8 = 2 has no solution.
Quick Check
11-3

24. Solving Radical Equations
ALGEBRA 1 LESSON 11-3
Solve each radical equation.
x – 3 = x – 2 = x + 2
x + 7 = 5x – x = 2x + 8
x = 3 2 5 7 2 5 4
no solution
11-3