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Algebra II : Solving Exponential Equations

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Example Question #101 : Solving Exponential Equations

Solve: $2^{5x+2} = (\frac{1}{2})^{10x+3}$

Possible Answers:

$\frac{1}{15}$

$\frac{1}{12}$

$-\frac{1}{6}$

$-\frac{1}{3}$

$-\frac{1}{9}$

Correct answer:

$-\frac{1}{3}$

Explanation:

Change the base of the right side.

$\frac{1}{2} =2^{-1}$

Rewrite the base of the right side.

$2^{5x+2} = (2^{-1})^{10x+3}$

Set the exponents equal to each other now that both of the bases are alike.

5x+2 = -1(10x+3)

Simplify both sides.

5x+2 = -10x-3

Add 10x on both sides.

5x+2 +10x= -10x-3+10x

15x+2= -3

Subtract two on both sides.

15x+2-2= -3-2

15x=-5

Divide by 15 on both sides.

$\frac{15x}{15}=\frac{-5}{15}$

Reduce both fractions.

The answer is: $-\frac{1}{3}$

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Example Question #102 : Solving Exponential Equations

Solve the equation: $8^{10x} = 8^{-9x-19}$

Possible Answers:

$\textup{There is no solution.}$

$\frac{19}{8}$

Correct answer:

Explanation:

The bases on both sides of the equation are equal. This means we can set the powers equal to each other.

10x= -9x-19

Add on both sides.

10x+9x= -9x-19+9x

19x = -19

Divide by 19 on both sides.

$\frac{19x}{19} = \frac{-19}{19}$

The answer is:

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Example Question #103 : Solving Exponential Equations

Solve the equation: $4^{3x+2} = (\frac{1}{4})^{-9x+3}$

Possible Answers:

$\textup{No solution.}$

$\frac{5}{3}$

$-\frac{5}{12}$

$\frac{5}{12}$

$\frac{5}{6}$

Correct answer:

$\frac{5}{6}$

Explanation:

Change the base on the right side to match the left side.

$\frac{1}{4} = 4^{-1}$

$4^{3x+2} = (4)^{-1(-9x+3)}$

Set the exponents equal to each other since both bases are equal.

3x+2 = -1(-9x+3)

Simplify both sides.

3x+2 =9x-3

Subtract both sides by and add both sides by three.

3x+2 -(3x)+[3]=9x-3-(3x)+[3]

5=6x

Divide both sides by six.

$\frac{5}{6}=\frac{6x}{6}$

The answer is: $\frac{5}{6}$

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Example Question #104 : Solving Exponential Equations

Solve: $16^{26} = 8^{3x+3}$

Possible Answers:

$\frac{95}{9}$

$\textup{The answer cannot be determined.}$

$\frac{35}{9}$

$\frac{17}{3}$

$\frac{35}{3}$

Correct answer:

$\frac{95}{9}$

Explanation:

In order to solve the equation, we will need to change the bases of both sides.

16= 2^4

8=2^3

Rewrite the equation using these two bases.

$(2^4)^{26} =(2^3)^{3x+3}$

Now that both sides have equal bases, we can set the exponents alike.

4(26) = 3(3x+3)

Simplify both sides.

104 = 9x+9

Subtract nine from both sides, and then divide both sides by nine.

104-9 = 9x+9-9

95=9x

$\frac{95}{9}=\frac{9x}{9}$

The answer is: $\frac{95}{9}$

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Example Question #105 : Solving Exponential Equations

Solve: $2^6 = (\frac{1}{8})^{3x+3}$

Possible Answers:

$-\frac{17}{3}$

$-\frac{5}{3}$

$-\frac{5}{8}$

$\frac{55}{9}$

$\frac{1}{3}$

Correct answer:

$-\frac{5}{3}$

Explanation:

Change the base of the fraction to base 2.

$\frac{1}{8} =2^{-3}$

$2^6 = (2^{-3})^{3x+3}$

Now that both bases are similar, we can set the powers equal to each other.

6 = -3(3x+3)

Use distribution to simplify the right side.

6=-9x-9

Add 9 on both sides.

6+9=-9x-9+9

15=-9x

Divide by negative nine on both sides.

$\frac{15}{-9}=\frac{-9x}{-9}$

$-\frac{15}{9}$

Reduce this fraction.

The answer is: $-\frac{5}{3}$

Report an Error

Example Question #106 : Solving Exponential Equations

Evaluate: $2^{8x+6} = 16^{30}$

Possible Answers:

$\frac{11}{5}$

$\frac{13}{4}$

$\frac{57}{4}$

$\frac{71}{4}$

Correct answer:

$\frac{57}{4}$

Explanation:

Change the base of the second number to base two.

16= 2^4

We can then replace the term and use the power rule of exponents to simplify the equation.

$2^{8x+6} = 2^{4(30)}$

Set the powers equal now that we have same bases.

8x+6 = 120

Subtract 6 from both sides.

8x+6 -6= 120-6

8x= 114

Divide by eight on both sides.

$\frac{8x}{8}= \frac{114}{8}$

Reduce the fractions.

The answer is: $\frac{57}{4}$

Report an Error

Example Question #107 : Solving Exponential Equations

Evaluate: $e= (\frac{1}{e^2})^{3x-3}$

Possible Answers:

$\frac{2}{3}$

$\textup{No solution.}$

$\frac{5}{6}$

Correct answer:

$\frac{5}{6}$

Explanation:

Convert the bases to to some power on both sides.

e= e^1

$\frac{1}{e^2} = e^{-2}$

Rewrite the terms of the equation.

$e^1= (e^{-2})^{3x-3}$

With similar bases, we can set the powers equal to each other.

1=(-2)(3x-3)

Simplify the right side by distribution.

1=-6x+6

Subtract six on both sides, and divide by negative six.

1-6=-6x+6-6

5=-6x

$\frac{5}{-6}=\frac{-6x}{-6}$

The answer is: $\frac{5}{6}$

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Example Question #108 : Solving Exponential Equations

Solve: $(2x-3)^{\frac{1}{4}} =3$

Possible Answers:

$\textup{No solution.}$

Correct answer:

Explanation:

We can rewrite the fractional exponent as a radical.

$\sqrt[4]{2x-3}=3$

Raise both sides by the power of four to eliminate the radical.

$(\sqrt[4]{2x-3} )^4=3^4$

2x-3 = 81

Add three on both sides.

2x-3+3 = 81+3

2x=84

Divide by two on both sides.

$\frac{2x}{2}=\frac{84}{2}$

Reduce both fractions.

The answer is:

Report an Error

Example Question #109 : Solving Exponential Equations

Solve: $\frac{1}{3} = 81^{-3x+5}$

Possible Answers:

$\frac{7}{3}$

$-\frac{19}{12}$

$\frac{7}{4}$

$\frac{7}{6}$

$\frac{19}{12}$

Correct answer:

$\frac{7}{4}$

Explanation:

To be able to solve this equation, we will need to change the bases on both sides to a common base.

Choose three as the common base.

$\frac{1}{3} =3^{-1}$

81=3^4

Replace the terms into the equation.

$3^{-1} =3^{4(-3x+5)}$

With the common bases, we can set the powers equal to each other.

-1 = 4(-3x+5)

Distribute the four on the right side.

-1 = -12x+20

Subtract 20 on both sides.

-1-20 = -12x+20-20

-21 = -12x

Divide by negative 12 on both sides.

$\frac{-21 }{-12}= \frac{-12x}{12}$

Reduce both fractions.

$\frac{-21 }{-12} = \frac{7\times 3}{4\times 3}$

The answer is: $\frac{7}{4}$

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Example Question #110 : Solving Exponential Equations

Solve: $4^{x+10} = (\sqrt2)^{-2x+3}$

Possible Answers:

$-\frac{17}{4}$

$-\frac{37}{6}$

$-\frac{23}{4}$

$\textup{No solution.}$

$-\frac{43}{6}$

Correct answer:

$-\frac{37}{6}$

Explanation:

Convert the bases to a common base so that the exponential powers can be set equal to each other. We can choose base two and rewrite both sides using exponential powers.

The radical is to the power of one-half.

$(2^2)^{x+10} = (2^{\frac{1}{2}})^{(-2x+3)}$

Set the powers equal to each other.

$2(x+10) =\frac{1}{2}(-2x+3)$

Distribute the outer terms into the binomials.

$2x+20 = -x+\frac{3}{2}$

Multiply by two on both sides to eliminate the fraction.

$2(2x+20) =2( -x+\frac{3}{2})$

4x+40 = -2x+3

Add on both sides.

4x+40+2x = -2x+3+2x

6x+40 = 3

Subtract 40 on both sides.

6x+40-40 = 3-40

6x=-37

Divide by 6 on both sides.

$\frac{6x}{6}=\frac{-37}{6}$

The answer is: $-\frac{37}{6}$

Report an Error

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Algebra II : Solving Exponential Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #101 : Solving Exponential Equations

Example Question #102 : Solving Exponential Equations

Example Question #103 : Solving Exponential Equations

Example Question #104 : Solving Exponential Equations

Example Question #105 : Solving Exponential Equations

Example Question #106 : Solving Exponential Equations

Example Question #107 : Solving Exponential Equations

Example Question #108 : Solving Exponential Equations

Example Question #109 : Solving Exponential Equations

Example Question #110 : Solving Exponential Equations

All Algebra II Resources

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