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High School Math : Solving Exponential Equations

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Example Questions

Example Question #21 : Solving And Graphing Exponential Equations

$y=\frac{4x^2-2}{2x^3-27}$

What are the x-intercepts of the equation?

Possible Answers:

$x=\sqrt{2}, -\sqrt{2}$

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

$x=\frac{1}{\sqrt{2}}$

$x=\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}$

$x=\sqrt{2}$

Correct answer:

$x=\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}$

Explanation:

To find the x-intercepts, set the numerator equal to zero and solve.

0=4x^2-2

2=4x^2

$\frac{2}{4}=x^2$

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

$\sqrt{\frac{1}{2}}=\sqrt{x^2}$

We can simplify from here:

$\frac{\sqrt1}{\sqrt2}}=\sqrt{x^2}$

$\frac{1}{\sqrt2}}=\sqrt{x^2}$

Now we need to rationalize. Because we have a square root on the bottom, we need to get rid of it. Since $2=\sqrt{2}^2$ , we can multiply $\frac{1}{\sqrt2}*\frac{\sqrt{2}}{\sqrt{2}}$ to get rid of the radical in the denominator.

$\frac{1}{\sqrt2}*\frac{\sqrt{2}}{\sqrt{2}}=x$

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

Since we took a square root, remember that our answer can be either positive or negative, as a positive squared is positive and a negative squared is also positive.

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Example Question #22 : Solving And Graphing Exponential Equations

$y=\frac{x-3}{x^2-12}$

What are the y-intercepts of this equation?

Possible Answers:

There are no y-intercepts.

$y=\frac{1}{4}$

$y=2\sqrt{3}$

y=0

y=4

Correct answer:

$y=\frac{1}{4}$

Explanation:

To find the y-intercept, set x=0 and solve.

$y=\frac{x-3}{x^2-12}$

$y=\frac{(0)-3}{(0)^2-12}$

$y=\frac{-3}{-12}$

$y=\frac{1}{4}$

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Example Question #42 : Exponents

$y=\frac{x^2-64}{x+2}$

What are the y-intercepts of this equation?

Possible Answers:

y=-32

There are no y-intercepts for the equation.

y=0

y=32

y=-2

Correct answer:

y=-32

Explanation:

To find the y-intercept, set x=0 and solve.

$y=\frac{x^2-64}{x+2}$

$y=\frac{(0)^2-64}{0+2}$

$y=\frac{-64}{2}$

y=-32

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Example Question #43 : Exponents

$y=\frac{3x^2-5}{2x^2+7}$

What are the x-intercepts of the equation?

Possible Answers:

$x=\sqrt{\frac{5}{3}}$

x=0

There are no horizontal asymptotes.

x=1

$x=\sqrt{\frac{5}{3}}, -\sqrt{\frac{5}{3}}$

Correct answer:

$x=\sqrt{\frac{5}{3}}, -\sqrt{\frac{5}{3}}$

Explanation:

To find the x-intercepts, we set the numerator equal to zero and solve.

0=3x^2-5

5=3x^2

$\frac{5}{3}=x^2$

$\sqrt{\frac{5}{3}}=\sqrt{x^2}$

$\sqrt{\frac{5}{3}}=x$

However, the square root of a number can be both positive and negative.

Therefore the roots will be $x=\sqrt{\frac{5}{3}}, -\sqrt{\frac{5}{3}}.$

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Example Question #44 : Exponents

$y=\frac{x^2-64}{x+2}$

What are the x-intercepts of the equation?

Possible Answers:

x=0

There are no real x-intercepts.

x=8, -8

There are no x-intercepts.

x=-2

Correct answer:

x=8, -8

Explanation:

To find the x-intercepts, set the numerator equal to zero.

0=x^2-64

64=x^2

$\sqrt{64}=\sqrt{x^2}$

8,-8=x

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Example Question #1 : Solving Exponential Equations

Solve the equation for .

$\small 9^x=3^6$

Possible Answers:

$\small x=2$

$\small x=1$

$\small x=0$

$\small x=3$

Correct answer:

$\small x=3$

Explanation:

Begin by recognizing that both sides of the equation have a root term of .

$\small 9^x=3^6$

(3^2)^x=3^6

Using the power rule, we can set the exponents equal to each other.

$3^{(2*x)}=3^6$

$\small 2x=6$

$\small x=3$

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Example Question #141 : Algebra Ii

The population of a certain bacteria increases exponentially according to the following equation:

$P(t)=2000e^{^{2t}}$

where P represents the total population and t represents time in minutes.

How many minutes does it take for the bacteria's population to reach 48,000?

Possible Answers:

$\frac{\log 24}{2}$

$\frac{2}{\ln 24}$

$\frac{\ln24}{2}$

$\ln 12$

$\frac{2}{\log 24}$

Correct answer:

$\frac{\ln24}{2}$

Explanation:

The question gives us P (48,000) and asks us to find t (time). We can substitute for P and start to solve for t:

$48,000 = 2000e^{2t}$

$24 = e^{2t}$

Now we have to isolate t by taking the natural log of both sides:

$\ln 24 = \ln e^{2t}$

$\ln 24 = (2t)\ln e$

And since $\ln e = 1$ , t can easily be isolated:

$\ln 24 = 2t$

$\frac{\ln24}{2} = t$

Note: $\frac{\ln 24}{2}$ does not equal $\ln 12$ . You have to perform the log operation first before dividing.

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Example Question #1 : Solving And Graphing Exponential Equations

Solve the equation for .

$\small 3^{2x}=81$

Possible Answers:

x=3

x=1

x=9

x=2

x=4

Correct answer:

x=2

Explanation:

Begin by recognizing that both sides of the equation have the same root term, .

$\small 3^{2x}=81$

$\small 3^{2x}=9^2$

$3^{2x}=(3^2)^2$

We can use the power rule to combine exponents.

$3^{2x}=3^4$

Set the exponents equal to each other.

2x=4

$\small x=2$

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Example Question #131 : Mathematical Relationships And Basic Graphs

Solve for :

x^3 - 9x =0

Possible Answers:

x = 0

$x = \frac{1}{3}$

$x = \{-3,0,3\}$

$x = \{-3,3\}$

x = 3

Correct answer:

$x = \{-3,0,3\}$

Explanation:

Pull an out of the left side of the equation.

$\rightarrow x(x^2 - 9) =0$

Use the difference of squares technique to factor the expression in parentheses.

$\rightarrow x(x+3)(x-3) =0$

Any number that causes one of the terms , x+3 , or x-3 to equal is a solution to the equation. These are , , and , respectively.

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High School Math : Solving Exponential Equations

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #21 : Solving And Graphing Exponential Equations

Example Question #22 : Solving And Graphing Exponential Equations

Example Question #42 : Exponents

Example Question #43 : Exponents

Example Question #44 : Exponents

Example Question #1 : Solving Exponential Equations

Example Question #141 : Algebra Ii

Example Question #1 : Solving And Graphing Exponential Equations

Example Question #131 : Mathematical Relationships And Basic Graphs

All High School Math Resources

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