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High School Math : High School Math

Study concepts, example questions & explanations for High School Math

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8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Simplifying Exponents

$\log_{x}27e^{3}=3$

Solve for .

Possible Answers:

1/2e

Correct answer:

Explanation:

First, set up the equation: x^3=27 e^3 . Simplifying this result gives x = 3 .

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Example Question #2 : Simplifying Exponents

What is the largest positive integer, , such that 2^z is a factor of 16^4 ?

Possible Answers:

Correct answer:

Explanation:

16^4 = (2^4)^4 = 2^1^6 . Thus, is equal to 16.

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Example Question #1 : Simplifying Exponents

Order the following from least to greatest:

$25^{100}$

$2^{300}$

$3^{500}$

$4^{400}$

$2^{600}$

Possible Answers:

$2^{300}, 2^{600}, 3^{500}, 25^{100}, 4^{400}$

$2^{600}, 2^{300}, 25^{100}, 3^{500}, 4^{400}$

$2^{300}, 25^{100}, 2^{600}, 3^{500}, 4^{400}$

$25^{100}, 2^{300}, 2^{600}, 3^{500}, 4^{400}$

$3^{500}, 4^{400}, 2^{300}, 25^{100}, 2^{600}$

Correct answer:

$2^{300}, 25^{100}, 2^{600}, 3^{500}, 4^{400}$

Explanation:

In order to solve this problem, each of the answer choices needs to be simplified.

Instead of simplifying completely, make all terms into a form such that they have 100 as the exponent. Then they can be easily compared.

2^3^0^0 = (2^3)^1^0^0 = 8^1^0^0 , 3^5^0^0 = (3^5)^1^0^0 = 243^1^0^0 , 4^4^0^0 = (4^4)^1^0^0 = 256^1^0^0 , and 2^6^0^0 = (2^6)^1^0^0 = 64^1^0^0 .

Thus, ordering from least to greatest: 8^1^0^0, 25^1^0^0, 64^1^0^0, 243^1^0^0, 256^1^0^0 .

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Example Question #1 : Review And Other Topics

Simplify the expression:

$(3x^4y^2)(4xy^2)^{-3}$

Possible Answers:

$\frac{3x}{16y^3}$

Cannot be simplified

$\frac{3x}{64y^4}$

$\frac{3x}{64y^3}$

Correct answer:

$\frac{3x}{64y^4}$

Explanation:

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

$(3x^{4}y^2)(4xy^2)^{-3}= (3xy^2)(4^{-3}x^{-3}y^{-6})$

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

$\frac{3x^4y^2}{64x^3y^6}$

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

$\frac{3x^4y^2}{64x^3y^6}=\frac{3}{64}x^{4-3}y^{2-6}=\frac{3}{64}xy^{-4}$

$\frac{3x}{64y^4}$

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Example Question #1 : Solving And Graphing Exponential Equations

$y=\frac{x^2-9}{x^2-16}$

What are the y-intercepts of this equation?

Possible Answers:

$y=\frac{16}{9}$

y=0

$y=\frac{9}{16}$

y=4,-4

y=3,-3

Correct answer:

$y=\frac{9}{16}$

Explanation:

To find the y-intercepts, set the value equal to and solve.

$y=\frac{x^2-9}{x^2-16}$

$y=\frac{(0)^2-9}{(0)^2-16}$

$y=\frac{-9}{-16}$

$y=\frac{9}{16}$

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Example Question #1 : Understanding Asymptotes

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

What are the horizontal asymptotes of this equation?

Possible Answers:

There are no horizontal asymptotes.

y=2

y=1

$y=\frac{1}{2}$

y=0

Correct answer:

y=0

Explanation:

When looking for the horizontal asymptotes, examine the exponents of the variables. Because the variable in the denominator has a higher exponent than the variable in the numerator, the horizontal asymptote will be at y=0 .

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Example Question #1 : Solving And Graphing Exponential Equations

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

What are the vertical asymptotes of the equation?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To find the vertical asymptotes, set the denominator equal to zero and solve.

0=2x^3-27

27=2x^3

$\frac{27}{2}=x^3$

$\sqrt[3]{\frac{27}{2}}=\sqrt[3]{x^3}$

${\frac{\sqrt[3]{27}}{\sqrt[3]2}}=\sqrt[3]{x^3}$

$\frac{3}{\sqrt[3]{2}}=x$

However, we need to rationalize from here. We need to get rid of the cubed root in the denominator.

$(\sqrt[3]{2})^3=\sqrt[3]{2}*\sqrt[3]{2}*\sqrt[3]{2}$ .

Therefore:

$\frac{3}{\sqrt[3]{2}}*\frac{\sqrt[3]{2}}{\sqrt[3]{2}}*\frac{\sqrt[3]{2}}{\sqrt[3]{2}}=x$

$\frac{3*(\sqrt[3]{2})^2}{(\sqrt[3]{2})^3}=x$

Bring the exponent from the numerator under the radical:

$\frac{3\sqrt[3]{2^2}}{2}=x$

Simplify:

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

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Example Question #2 : Solving And Graphing Exponential Equations

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

What is the horizontal asymptote of this equation?

Possible Answers:

There is no horizontal asymptote.

y=0

y=1

$y=\sqrt{12}$

y=3

Correct answer:

y=0

Explanation:

To find the horizontal asymptotes, we compare the exponents of in our fraction. Because the denominator variable's exponent is greater than the numerator variable's exponent, our horizontal asymptote is at y=0 .

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Example Question #3 : Understanding Asymptotes

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

What are the vertical asymptotes of the equation?

Possible Answers:

There are no vertical asymptotes.

x=3

x=0

$x=2\sqrt{3},-2\sqrt{3}$

$x=\sqrt{12}$

Correct answer:

$x=2\sqrt{3},-2\sqrt{3}$

Explanation:

To find the vertical asymptotes, we set the denominator equal to zero.

0=x^2-12

12=x^2

$\sqrt{12}=\sqrt{x^2}$

$\sqrt{4*3}=x$

Because the square root only gives us the absolute value, our answer will be:

$x=2\sqrt{3}, -2\sqrt{3}$

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Example Question #4 : Understanding Asymptotes

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

What are the horizontal asymptotes of this equation?

Possible Answers:

There are no horizontal asymptotes.

y=0

$y=\frac{2}{3}$

$y=\frac{3}{2}$

y=1

Correct answer:

$y=\frac{3}{2}$

Explanation:

Since the exponents of the variables in both the numerator and denominator are equal, the horizontal asymptote will be the coefficient of the numerator's variable divided by the coefficient of the denominator's variable.

For this problem, since we have $\frac{3x^2}{2x^2}$ , our asymptote will be $y=\frac{3}{2}$ .

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High School Math : High School Math

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #2 : Simplifying Exponents

Example Question #2 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Review And Other Topics

Example Question #1 : Solving And Graphing Exponential Equations

Example Question #1 : Understanding Asymptotes

Example Question #1 : Solving And Graphing Exponential Equations

Example Question #2 : Solving And Graphing Exponential Equations

Example Question #3 : Understanding Asymptotes

Example Question #4 : Understanding Asymptotes

All High School Math Resources

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