Warm-Up Exercises
Evaluate the expression.
1. 64
2
3
ANSWER
2. (–32)
16
3
5
ANSWER
–8
Warm-Up Exercises
3.6 Solve Radical Equations Day 2
• How do you solve equations with fraction
exponents?
• How do you solve equations with two radicals?
EXAMPLE
Warm-Up4Exercises
Solve an equation with a rational exponent
Solve (x + 2)3/4 – 1 = 7.
(x + 2)3/4 – 1 = 7
(x + 2)3/4 = 8
Write original equation.
Add 1 to each side.
3/4 4/3
Raise each side to the power 4 .
3
x + 2 = (8 1/3)4
Apply properties of exponents.
x + 2 = 24
Simplify.
x + 2 = 16
Simplify.
(x + 2)
= 8 4/3
x = 14
Subtract 2 from each side.
EXAMPLE
Warm-Up4Exercises
Solve an equation with a rational exponent
ANSWER
The solution is 14. Check this in the original
equation.
So plug in 14 for x in the equation: (x + 2)3/4 – 1 = 7
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Solve the equation. Check your solution.
5.
3x3/2 = 375
3x3/2 = 375
x3/2 = 125
Write original equation.
Divide each side by 3.
2
3
(x3/2)2/3 = (125)2/3 Raise each side to the power .
x = 25
Simplify.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Solve the equation. Check your solution.
7.
– 2 x1/5 = –2
3
– 2 x1/5 = –2
3
x1/5= 3
(x1/5 )5= 35
x = 243
Write original equation.
Divide each side by –2/3.
Raise each side to the power 5.
Simplify.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Solve the equation. Check your solution.
9. (x – 5)4/3 = 81
(x – 5)4/3 = 81
[(x – 5)4/3]3/4 = (81)3/4
Write original equation.
Raise each side to the power 3/4.
x – 5 = (811/4)3
Apply properties of exponent.
x – 5 = ±33
Simplify.
x – 5 = ±27
Simplify.
x – 5 = 27 or x – 5 = 27
x = 32 or x = –22
Let (x – 5) equal 27 and –27.
Add 5 to both sides of
each equation.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
Solve the equation. Check your solution.
10.
(x + 2)2/3 + 3 = 7
(x + 2)2/3 +3 = 7
Write original equation.
(x + 2)2/3 = 4
Subtract each side by 3.
[(x + 2)2/3]3/2 = 43/2
x + 2 = (41/2)3
x +2 = 8 or x + 2 = −8
x = −10 or 6
Raise each side to the power 3/2.
Apply properties of exponent.
Simplify.
Simplify.
EXAMPLE
Warm-Up6Exercises
Solve an equation with two radicals
Solve √x + 2 + 1 = √ 3 – x .
SOLUTION
√x + 2 + 1 = √ 3 – x
2
√x + 2 + 1 = √ 3 – x
x + 2 +2 √ x + 2 +1 = 3 – x
2√ x + 2 = – 2x
Write original equation.
2
Square each side.
Expand left side and simplify
right side.
Isolate radical expression.
EXAMPLE
Warm-Up6Exercises
Solve an equation with two radicals
√ x + 2 = –x
Divide each side by 2.
2
√ x + 2 = ( –x)2
x + 2 = x2
Square each side again.
Simplify.
0 = x2 – x – 2
0 = (x – 2)(x + 1)
Write in standard form.
Factor.
x–2=0
or
x+1=0
Zero-product property.
x=2
or
x = –1
Solve for x.
EXAMPLE
Warm-Up6Exercises
Solve an equation with two radicals
Check x = 2 in the
original equation.
√x + 2 +1 = √ 3 – x
√2 + 2 +1 =? √ 3 – 2
√ 4 +1 =? √ 1
3 =/ –1
Check x = – 1 in the
original equation.
√x + 2 +1 = √ 3 – x
√–1 + 2 +1 =? √ 3 – (–1)
√ 1 +1 =? √ 4
2=2
ANSWER
The only solution is −1.
(The apparent solution 2 is extraneous.)
EXAMPLE
Warm-Up6Exercises
Solve an equation with two radicals
METHOD 2
Use: a graph to solve the equation. Use a graphing
calculator to graph y1 = x + 2 + 1 and y2 = 3 – x . Then
find the intersection points of the two graphs by using
the intersect feature. You will find that the only point
of intersection is (−1, 2). Therefore, −1 is the only
solution of the equation
x+2 +1 = 3–x
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 5 and 6
Solve the equation. Check for extraneous solutions
12. 10x + 9 = x + 3
10x + 9 = x + 3
(
2
)
10x + 9 =? (x + 3)2
Write original equation.
Square each side.
10x + 9 = x + 6x +9
Expand right side and simplify
left side.
x2 – 4x = 0
Write in standard form.
2
x (x – 4) = 0
Factor.
(x – 4) = 0 or x = 0 Zero-product property.
x = 4 or x = 0 Solve for x.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 5 and 6
Check x = 4 in the
original equation.
Check x = 0 in the
original equation.
10x + 9 = x + 3
10x + 9 = x + 3
10x 4+ 9 =? 4 + 3
10x 0+ 9 =? 0 + 3
40 + 9 =? 7
49 =? 7
7=7
The solution are 4 and 0.
9 =? 3
3=3
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 5 and 6
Solve the equation. Check for extraneous solutions
13. 2x + 5 =
x+7
2x + 5 =
x+7
(
2x + 5
)(
)
2
2
=
Write original equation.
x+ 7
Square each side.
2x + 5 = x + 7
Simplify both the sides.
x –2 =0
Simplify.
x=2
Simplify.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 5 and 6
Check x = 2 in the original equation
2x + 5 =
x+7
2.2 + 5 =?
2+7
9
=?
9
3=3
The solution is 2.
Warm-Up Exercises
•
How do you solve equations with fraction exponents?
Multiply by the reciprocal.
•
How do you solve equations with two radicals?
Make sure you have only one radical on each side of the
equation and raise both sides of the equation to the same
power.
Warm-Up Exercises
Hw 3.6
Page 208
#5, 6-12 even, 13,
14-18 even, 25-28,
35-37, 45-51 odd
BONUS: 53, 60