Unit 6GA2Test Review

Find the indicated real nth root(s) of a.

a. n = 3, a = –216 b. n = 4, a = 81

SOLUTION

b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 = ±3

a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3 = –216, you can write = 3√ –216 = –6 or (–216)1/3 = –6.

Evaluate:

SOLUTIONRational Exponent Form Radical Form

a. 163/2 (161/2)3 = 43= 64=

b. 32–3/5 = 1323/5 = 1

(321/5)3

= 123

18

=

a. 163/2 ( )3 = 16 43= 64=

b. 32–3/51

323/5 = 1( )35 32

=

= 123

18

=

(a) 163/2 (b) 32–3/5

Solve the equation.

4x5 = 128

x5 32=

x = 325

x 2=

SOLUTION

( x + 5 )4 = 16

SOLUTION

( x + 5 )4 = 16

( x + 5 ) = + 4 16

x = + 4 16 – 5x = 2 – 5 or x = – 2 – 5

x = – 3 or x = –7

Solve the equation.

1. (51/3 71/4)3 = (51/3)3 (71/4)3

2. 23/4 21/2

= 3(1 – 1/4)

Simplify the expressions.

= 205

1/2 3(41/2)3=

51/3 3 71/4 3= 51 73/4= 5 73/4=

25/4=2(3/4 + 1/2)=

31

31/4= 33/4=

(22)3/2= 8=

1 4

33.

331 2

1 2

204.

5

Simplify the expressions.

274 34 27 34= 814= = 3

= 23

533 23= 5=

23533 2

245

2=35

4585

85=

325245

=

4 4. 27 3a

3

3

250.

2b

53

. 4

c

Simplify the expression. Assume all variables are positive.

= 33(q3)33 333 (q3)33= = 3q3

=x105

y55(x2)55

=y55

=x2

y

= 2x(1 – 1/2)y(3/4 –1/2) = 2x1/2y1/4

93 27q

10

55

xy

3 4

1 2 1 2

63xyx y

a.

b.

c.

Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following, state the domain.

f (x) + g(x)1.

SOLUTION

f (x) + g(x) = –2x2/3 + 7x2/3 = (–2 + 7)x2/3 = 5x2/3

f (x) – g(x)2.

SOLUTION

f (x) – g(x) = –2x2/3 – 7x2/3 = [–2 + ( –7)]x2/3 = –9x2/3

Let f (x) = 3x and g(x) = x1/5. Find the following, state the domain.

SOLUTION

SOLUTION

f (x) g(x)a.

f (x) g(x) = 3x x1/5 = 3(x ) 1 + 1/5 = 3x6/5

f (x)g(x) =

3xx1/5 = 3(x ) 1 – 1/5 = 3x4/5

( )( )f xg x

b.

Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.

a g(f(5))

SOLUTION

To evaluate g(f(5)), you first must find f(5).

f(5)Then g( f(3))

= 3(5) – 8 = 7= g(7)

= 2(7)2

= 2(49)= 98

b f(g(5))

SOLUTION

To evaluate f(g(5)), you first must find g(5).

g (5)

Then f( g(5))

= 2(5)2 = 2(25)

= f(50)

= 3(50) – 8

= 150 – 8

= 142.

= 50

f(x) = –3x – 1

Find the inverse.

y = –3x + 1

x = –3y +1

x – 1 = –3y

x 13 = y

SOLUTION

ANSWER g–1(x) = 33√ x

Find the inverse.

31( )

27g x x

f(x) = –x3 + 4

ANSWER f –1(x) = 3√ 4 – x

Find the inverse.

Graph the function.

Then state the domain and range.

ANSWER

Domain : x > 0 ,range : y < 0.

4 2y x

Graph the function.

Then state the domain and range.

ANSWER

Domain :all real numbers,

range: all real numbers.

3( ) 2 3g x x

Solve (x + 2)3/4 – 1 = 7

(x + 2)3/4 – 1 = 7

(x + 2)3/4 = 8

(x + 2)3/4 4/3= 8 4/3

x + 2 = (8 1/3)4

x + 2 = 24

x + 2 = 16

x = 14

SOLUTION

Solve the equation. Check your solution.

3x3/2 = 375

3x3/2 = 375

x3/2 = 125

(x3/2)2/3 = (125)2/3

x = 25

SOLUTION

x + 1 = 7x + 15

(x + 1)2 = ( 7x + 15)2

x2 + 2x + 1 = 7x + 15

x2 – 5x – 14 = 0

(x – 7)(x + 2) = 0

x – 7 = 0 or x + 2 = 0

x = 7 or x = –2

SOLUTION

1 7 15x x Solve

x + 6 – 4 x + 6 + 4 = x – 2

– 4 x + 6 = – 12

x + 6 = 3

Solve the equation. Check for extraneous solutions.

6 2 2x x

x + 6 = 9

x = 3

SOLUTION