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High School Math : Exponents

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Example Questions

Example Question #41 : Exponents

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

What are the x-intercepts of the equation?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

$\displaystyle 0=4x^2-2$

2=4x^2 $\displaystyle 2=4x^2$

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

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

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

We can simplify from here:

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

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

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

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

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

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 #42 : Exponents

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

What are the y-intercepts of this equation?

Possible Answers:

y=0 $\displaystyle y=0$

y=4 $\displaystyle y=4$

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

$y=2\sqrt{3}$ $\displaystyle y=2\sqrt{3}$

There are no y-intercepts.

Correct answer:

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

Explanation:

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

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

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

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

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

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Example Question #123 : Mathematical Relationships And Basic Graphs

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

What are the y-intercepts of this equation?

Possible Answers:

There are no y-intercepts for the equation.

y=-2 $\displaystyle y=-2$

y=-32 $\displaystyle y=-32$

y=0 $\displaystyle y=0$

y=32 $\displaystyle y=32$

Correct answer:

y=-32 $\displaystyle y=-32$

Explanation:

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

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

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

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

y=-32 $\displaystyle y=-32$

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Example Question #12 : Solving Exponential Equations

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

What are the x-intercepts of the equation?

Possible Answers:

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

x=1 $\displaystyle x=1$

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

x=0 $\displaystyle x=0$

There are no horizontal asymptotes.

Correct answer:

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

Explanation:

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

$\displaystyle 0=3x^2-5$

5=3x^2 $\displaystyle 5=3x^2$

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

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

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

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

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

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Example Question #44 : Exponents

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

What are the x-intercepts of the equation?

Possible Answers:

There are no x-intercepts.

There are no real x-intercepts.

x=0 $\displaystyle x=0$

x=-2 $\displaystyle x=-2$

x=8, -8 $\displaystyle x=8, -8$

Correct answer:

x=8, -8 $\displaystyle x=8, -8$

Explanation:

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

$\displaystyle 0=x^2-64$

64=x^2 $\displaystyle 64=x^2$

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

8,-8=x $\displaystyle 8,-8=x$

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Example Question #1 : Solving Exponential Equations

Solve the equation for $\displaystyle x$.

$\small 9^x=3^6$ $\displaystyle \small 9^x=3^6$

Possible Answers:

$\small x=1$ $\displaystyle \small x=1$

$\small x=3$ $\displaystyle \small x=3$

$\small x=2$ $\displaystyle \small x=2$

$\small x=0$ $\displaystyle \small x=0$

Correct answer:

$\small x=3$ $\displaystyle \small x=3$

Explanation:

Begin by recognizing that both sides of the equation have a root term of $\displaystyle 3$.

$\small 9^x=3^6$ $\displaystyle \small 9^x=3^6$

$\displaystyle (3^2)^x=3^6$

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

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

$\small 2x=6$ $\displaystyle \small 2x=6$

$\small x=3$ $\displaystyle \small x=3$

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Example Question #141 : Algebra Ii

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

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

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:

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

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

$\frac{\ln24}{2}$ $\displaystyle \frac{\ln24}{2}$

$\ln 12$ $\displaystyle \ln 12$

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

Correct answer:

$\frac{\ln24}{2}$ $\displaystyle \frac{\ln24}{2}$

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:

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

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

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

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

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

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

$\ln 24 = 2t$ $\displaystyle \ln 24 = 2t$

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

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

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Example Question #2 : Solving Exponential Equations

Solve the equation for $\displaystyle x$.

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

Possible Answers:

x=2 $\displaystyle x=2$

x=1 $\displaystyle x=1$

x=9 $\displaystyle x=9$

x=3 $\displaystyle x=3$

x=4 $\displaystyle x=4$

Correct answer:

x=2 $\displaystyle x=2$

Explanation:

Begin by recognizing that both sides of the equation have the same root term, $\displaystyle 3$.

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

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

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

We can use the power rule to combine exponents.

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

Set the exponents equal to each other.

2x=4 $\displaystyle 2x=4$

$\small x=2$ $\displaystyle \small x=2$

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Example Question #43 : Exponents

Solve for $\displaystyle x$:

$\displaystyle x^3 - 9x =0$

Possible Answers:

x = 0 $\displaystyle x = 0$

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

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

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

x = 3 $\displaystyle x = 3$

Correct answer:

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

Explanation:

Pull an $\displaystyle x$ out of the left side of the equation.

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

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

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

Any number that causes one of the terms $\displaystyle x$, x+3 $\displaystyle x+3$, or x-3 $\displaystyle x-3$ to equal $\displaystyle 0$ is a solution to the equation. These are $\displaystyle 0$, $\displaystyle -3$, and $\displaystyle 3$, respectively.

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Example Question #1 : Graphing Exponential Functions

Find the $\displaystyle y$-intercept(s) of $y=\frac{x^3+2x+8}{x^2-36}$ $\displaystyle y=\frac{x^3+2x+8}{x^2-36}$.

Possible Answers:

y=6 $\displaystyle y=6$

$y=-\frac{2}{9}$ $\displaystyle y=-\frac{2}{9}$

$y=\frac{2}{9}$ $\displaystyle y=\frac{2}{9}$

This function does not cross the $\displaystyle y$-axis.

y=8 $\displaystyle y=8$

Correct answer:

$y=-\frac{2}{9}$ $\displaystyle y=-\frac{2}{9}$

Explanation:

To find the $\displaystyle y$-intercept, set x=0 $\displaystyle x=0$ in the equation and solve.

$y=\frac{x^3+2x+8}{x^2-36}$ $\displaystyle y=\frac{x^3+2x+8}{x^2-36}$

$y=\frac{(0)^3+2(0)+8}{(0)^2-36}$ $\displaystyle y=\frac{(0)^3+2(0)+8}{(0)^2-36}$

$y=\frac{(0)+(0)+8}{(0)-36}$ $\displaystyle y=\frac{(0)+(0)+8}{(0)-36}$

$y=\frac{8}{-36}$ $\displaystyle y=\frac{8}{-36}$

$y=-\frac{2}{9}$ $\displaystyle y=-\frac{2}{9}$

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High School Math : Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #41 : Exponents

Example Question #42 : Exponents

Example Question #123 : Mathematical Relationships And Basic Graphs

Example Question #12 : Solving Exponential Equations

Example Question #44 : Exponents

Example Question #1 : Solving Exponential Equations

Example Question #141 : Algebra Ii

Example Question #2 : Solving Exponential Equations

Example Question #43 : Exponents

Example Question #1 : Graphing Exponential Functions

All High School Math Resources

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