Simplify 243. Simplifying Radicals ALGEBRA 1 LESSON 11-1 = 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. Quick Check 11-1
Jan 03, 2016
Simplify 243.
Simplifying RadicalsSimplifying RadicalsALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
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.
Quick Check
11-1
Simplify 28x7.
Simplifying RadicalsSimplifying RadicalsALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
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.
11-1
Quick Check
Simplifying RadicalsSimplifying Radicals
Simplify each radical expression.
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
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.
11-1
Simplifying RadicalsSimplifying Radicals
(continued)
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
b. 7 5x • 3 8x
= 42x 10 Simplify.
= 21 • 2x 10 Simplify 4x2.
= 21 4x2 • 10 Use the Multiplication Property of
Square Roots.
= 21 4x2 • 10 4x2 is a perfect square and a
factor of 40x2.
7 5x • 3 8x = 21 40x2 Multiply the whole numbers and
use the Multiplication Property of
Square Roots.
11-1
Quick Check
Simplifying RadicalsSimplifying Radicals
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 LESSON 11-1ALGEBRA 1 LESSON 11-1
d = 1.5h
To the nearest mile, the distance you can see is 9 miles.
= 78 Multiply.
= 1.5 • 52 Substitute 52 for h.
8.83176 Use a calculator.
11-1
Quick Check
Simplifying RadicalsSimplifying Radicals
Simplify each radical expression.
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
= Simplify 64. 13
8
a. 1364
b. 49x4
7
x2 = Simplify 49 and x4.
= Use the Division Property of Square Roots.1364
13
64
= Use the Division Property of Square Roots.49x4
49
x4
11-1
Quick Check
Simplifying RadicalsSimplifying Radicals
= 12 Divide.120 10
= 4 • 3 4 is a perfect square and a factor of 12.
a. 120 10
Simplify each radical expression.
= 4 • 3 Use the Multiplication Property of Square Roots.
= 2 3 Simplify 4.
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
11-1
b. 75x5
48x
Simplifying RadicalsSimplifying Radicals
= Divide the numerator and denominator by 3x.75x5
48x25x4
16
= Use the Division Property of Square Roots.25x4
16
(continued)
= Use the Multiplication Property ofSquare Roots.
25 • x4
16
= Simplify 25, x4, and 16.5x2
4
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
11-1
Quick Check
3
7
3
7
7
7 7
7= • Multiply by to make the denominator a
perfect square.
Simplifying RadicalsSimplifying Radicals
Simplify each radical expression.
a. 3 7
= Simplify 49.3 7 7
= Use the Multiplication Property of Square Roots.3 7
49
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
11-1
= Simplify 36x4. 33x
6x2
Simplifying RadicalsSimplifying Radicals
(continued)
b. 11
12x3
Simplify the radical expression.
= • Multiply by to make the denominator a
perfect square.
3x
3x
3x
3x
11
12x3
11
12x3
= Use the Multiplication Property of Square Roots. 33x
36x4
ALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
11-1
Quick Check
Simplifying RadicalsSimplifying RadicalsALGEBRA 1 LESSON 11-1ALGEBRA 1 LESSON 11-1
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
11-1
Simplify 4 3 + 3.
Operations with Radical ExpressionsOperations with Radical ExpressionsALGEBRA 1 LESSON 11-2ALGEBRA 1 LESSON 11-2
= (4 + 1) 3 Use the Distributive Property to combine like radicals.
= 5 3 Simplify.
4 3 + 3 = 4 3 + 1 3 Both terms contain 3.
Quick Check
11-2
8 5 – 45 = 8 5 + 9 • 5 9 is a perfect square and a factor of 45.
Simplify 8 5 – 45.
Operations with Radical ExpressionsOperations with Radical ExpressionsALGEBRA 1 LESSON 11-2ALGEBRA 1 LESSON 11-2
= 8 5 – 9 • 5 Use the Multiplication Property of Square Roots.
= 8 5 – 3 5 Simplify 9.
= (8 – 3) 5 Use the Distributive Property tocombine like terms.
= 5 5 Simplify.
11-2
Quick Check
Operations with Radical ExpressionsOperations with Radical Expressions
Simplify 5( 8 + 9).
ALGEBRA 1 LESSON 11-2ALGEBRA 1 LESSON 11-2
5( 8 + 9) = 40 + 9 5 Use the Distributive Property.
= 4 • 10 + 9 5 Use the Multiplication Property of Square Roots.
= 2 10 + 9 5 Simplify.
11-2
Quick Check
Simplify ( 6 – 3 21)( 6 + 21).
Operations with Radical ExpressionsOperations with Radical ExpressionsALGEBRA 1 LESSON 11-2ALGEBRA 1 LESSON 11-2
( 6 – 3 21)( 6 + 21)
= 36 + 126 – 3 126 – 3 441 Use
FOIL.= 6 – 2 126 – 3(21) Combine like radicals and
simplify 36 and 441.
= 6 – 2 9 • 14 – 63 9 is a perfect square factor of 126.
= 6 – 2 9 • 14 – 63 Use the Multiplication Property of Square Roots.
= 6 – 6 14 – 63 Simplify 9.
= –57 – 6 14 Simplify.
11-2
Quick Check
Operations with Radical ExpressionsOperations with Radical ExpressionsALGEBRA 1 LESSON 11-2ALGEBRA 1 LESSON 11-2
= 2( 7 + 3) Divide 8 and 4 by the common factor 4.
= 2 7 + 2 3 Simplify the expression.
= Multiply in the denominator. 8( 7 + 3)
7 – 3
= Simplify the denominator. 8( 7 + 3)
4
Simplify . 8
7 – 3
= • Multiply the numerator and denominator by the conjugate of the denominator.
8
7 – 3
7 + 3
7 + 3
11-2
Quick Check
Solve each equation. Check your answers.
Solving Radical EquationsSolving Radical EquationsALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
a. x – 5 = 4
x = 9 Isolate the radical on the left side of the equation.
( x)2 = 92 Square each side.
x = 81
x – 5 = 4
4 = 4
Check: x – 5 = 4
81 – 5 4 Substitute 81 for x.
9 – 5 4
11-3
b. x – 5 = 4
Solving Radical EquationsSolving Radical EquationsALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
x – 5 = 16 Solve for x.
x = 21
( x – 5)2 = 42 Square each side.
(continued)
Check: x – 5 = 4 21– 5 = 4 Substitute 21 for x. 16 = 4
4 = 4
11-3
Quick Check
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.
Solving Radical EquationsSolving Radical EquationsALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
11-3
Solving Radical EquationsSolving Radical EquationsALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
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 8
(15)2 = ( h – 36)2 Square both sides.225 = h – 36261 = h
The hill is 261 ft high.
(continued)
11-3
Quick Check
Solve 3x – 4 = 2x + 3.
Solving Radical EquationsSolving Radical Equations
( 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.
The solution is 7.
Check: 3x – 4 = 2x + 3
3(7) – 4 2(7) + 3 Substitute 7 for x.
17 = 17
ALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
11-3
Quick Check
Solving Radical EquationsSolving Radical EquationsALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
(x)2 = ( x + 12)2 Square both sides.
x2 = x + 12
x2 – x – 12 = 0 Simplify.
The solution to the original equation is 4. The value –3 is an extraneous solution.
Solve x = x + 12.
(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
4 4 + 12 –3 –3 + 12
4 = 4 –3 = 3 /
11-3
Quick Check
Solving Radical EquationsSolving Radical EquationsALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
Solve 3x + 8 = 2.
3x = –6
( 3x)2 = (–6)2 Square both sides.
3x = 36
x = 12
3x + 8 = 2 has no solution.
Check: 3x + 8 = 2
3(12) + 8 2 Substitute 12 for x.
36 + 8 2
6 + 8 = 2 x = 12 does not solve the original equation./
11-3
Quick Check
Solving Radical EquationsSolving Radical Equations
Solve each radical equation.
1. 7x – 3 = 4 2. 3x – 2 = x + 2
3. 2x + 7 = 5x – 8 4. x = 2x + 8
5. 3x + 4 + 5 = 3
2
5 4
no solution
2 57
ALGEBRA 1 LESSON 11-3ALGEBRA 1 LESSON 11-3
11-3