Reviewing Algebra II: Solutions To Simplifying & Solving Exponential Equations

Solve each equation for the unknown and check:

 (1) 2a^-3 - 1 = 15

Solution:  Isolate exponent side:

2a^-3  = 16

Divide each side by 2:

a^-3  = 8

Get both sides to same base:

a^-3  = 8 =   2^-3

a =  2

Check in original eqn. 2a^-3 - 1 = 15

2(2)^-3 - 1 = 15 =  2 (8) - 1 = 15

16  - 1  = 15 

(2) 9^x =  55

Solution:  

Use the power rule: i.e. ln b^x   = x ln b, to bring the variable exponent (x) to the front, i.e.

x ln 9 =  ln 55

Solve by dividing both sides by ln 9:

x  =   ln 55/ ln 9

x =  4.007/ 2.197

x = 1.824

Check in original eqn.  x ln 9 =  ln 55

(1.824) ln 9  =   ln 55

(1.824) (2,197)  = 4.007  

3) 5x^1/2 + 7   = 22

Solution: Isolate exponent side:

5x^1/2    = 15

Divide both sides by 5:

x^1/2    = 3

Ö x  =   3

x =   9

Check in original eqn. 5x^1/2 + 7   = 22

5x^1/2 + 7   = 22

5 Ö 9   +  7  = 22

5 (3)  +  7  = 22

15 + 7  = 22 

4) 3a³   = 81

Solution:

Divide both sides by 3:

a³   = 27

Change both sides to same base, recognizing: 3 ³   = 27

Then: a³   =   3³   

a =  3

Check from original eqn.

3a³   = 81

3(3)³   = 81

3 (27)  = 81 

5) 4^x = 8 ^x+1

Solution:

Rem: same base exponents can be equated

Note: 4  = 2 ²     and:  8 = 2 ³

Rewrite eqn. for the same base:

2 ^2x = 2 ^3(x+1)

Equate exponents:

2x  =   3x  +3

Solve for x:  x  = -3

x  = -3

Check in original eqn.

4^(-3) = 8 ^(-3+1)

4^(-3) = 8 ^-2

0.16   =   0.16

6) b^-5 = 1/32

Solution:

1/b ⁵  =   1/32

1/b ⁵  =   1/2 ⁵  (Same base)

Þ   b  = 2 

 Check in original eqn.

2 ⁵ =   32  so:  2 ^-5  =    1/32

From exponent rule:     a ^-n  =  1/a ⁿ

 

7)  3 +   7 ^x-1  =  10

Solution:

Isolate exponent member:

 

 7 ^x-1  =  10  - 3  = 7

Use same base to set exps equal:

7 ^x-1  =    7

By rules of exponents:   a ¹  =   a

So:   7 =  7 ¹

Set exps equal:  x- 1 = 1  so x = 2

Check using original eqn.

 3 +   7 ^{2 -1}  =  10

3 +  7 ¹    =   10

3 +  7    =   10

For each of the following, write as a power with positive exponents in simplest form:

8)

Solution:  By properties of exponents:

 a ^{m/ n}     =   (a ^1/n ) ^m     = (a ^m) ^1/n

Þ

½ (18 a ⁶  b ²) ^1/2

=  ½ (3Ö2  a ³  b) =  3Ö2  a ³  b/ 2

9)

Solution:  By properties of exponents:

( ⁿ Ö a ) ^m     =   ⁿ Ö( a ^m )  =  (a ^1/n ) ^m 

Þ  

[9 ^1/5  ] ⁴ =  9 ^4/5   = 5.3

10)

Solution:  By properties of exponents:

( ⁿ Ö a ) ^m     =   ⁿ Ö( a ^m )  =  (a ^1/n ) ^m 

Þ

(w ¹⁵  x ²⁰ y ^-5 )  1/10

=  (w ^15/10  x ^20/10 y ^-5/10 )

 =  w ^3/2  x ² y ^-1/2