We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

High School Math : Pre-Calculus

Study concepts, example questions & explanations for High School Math

Create An Account

All High School Math Resources

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

Example Questions

← Previous 1 2 3 4 5 6 Next →

Example Question #2 : Simplifying Polynomial Functions

Simplify the following polynomial function:

$ab^{-2}(a^{2}b + a^{-1}b^{3}-a^{3}b^{-1})$

Possible Answers:

$\frac{a^{3}}{b}+b-\frac{a^{4}}{b^{3}}$

$\frac{a^{3}}{b}-\frac{a^{4}}{b^{3}}$

$\frac{a^{3}}{b}-b+\frac{a^{4}}{b^{3}}$

$\frac{a^{3}}{b}+b^{2}-\frac{a^{4}}{b^{3}}$

$\frac{a^{3}}{b}+\frac{a^{4}}{b^{3}}$

Correct answer:

$\frac{a^{3}}{b}+b-\frac{a^{4}}{b^{3}}$

Explanation:

First, multiply the outside term with each term within the parentheses:

$ab^{-2}(a^{2}b + a^{-1}b^{3}-a^{3}b^{-1})$

$a^{3}b^{-1} + b-a^{4}b^{-3}$

Rearranging the polynomial into fractional form, we get:

$\frac{a^{3}}{b}+b-\frac{a^{4}}{b^{3}}$

Report an Error

Example Question #1 : Exponential And Logarithmic Functions

You are given that $\log_{10} x = A$ and $\log_{10} y = B$ .

Which of the following is equal to $100^{2A-B}$ ?

Possible Answers:

$\frac{2x}{y}$

$\frac{x}{\sqrt{y}}$

$x^{4}y^{2}$

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

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

Correct answer:

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

Explanation:

Since $\log_{10} x = A$ and $\log_{10} y = B$ , it follows that $10^{A} = x$ and $10^{B} = y$

$100^{2A-B} = \left( 10^{2} \right ) ^{2A-B} =10^{2 (2A-B)} =10^{4A-2B}=\frac{10^{4A}}{10^{2B}} =\frac{\left ( 10^{A}\right )^4}{\left (10^{B}\right )^2}=\frac{x^4}{y^2}$

Report an Error

Example Question #21 : Pre Calculus

$\textup{Solve for }x\textup{: }\,\log_{2}32=x$

Possible Answers:

Correct answer:

Explanation:

$\textup{This can be re-written as an exponent problem: } 2^{x}=32$

$2\times2\times2\times2\times2=32\:\textup{ so }32=2^{5}\:\:\:x=5$

Report an Error

Example Question #2 : Exponential And Logarithmic Functions

What is $log_{2}(8)$ ?

Possible Answers:

$\frac{1}{8}$

Correct answer:

Explanation:

Recall that by definition a logarithm is the inverse of the exponential function. Thus, our logarithm corresponds to the value of in the equation:

$2^{x} = 8$ .

We know that $2^{3} = 8$ and thus our answer is .

Report an Error

Example Question #21 : Pre Calculus

Solve for : $\log_{2}\left ( x+4 \right ) + \log_{2}\left ( x+5 \right ) = \log_{2} 6$

Possible Answers:

$x = -7 \textrm{ or }x = -2$

$x = 1 \textrm{ or }x = \log_{2}7$

$x = 4 \textrm{ or }x = 128$

$x = 2 \textrm{ or }x = 7$

The correct solution set is not included among the other choices.

Correct answer:

The correct solution set is not included among the other choices.

Explanation:

$\log_{2}\left ( x+4 \right ) + \log_{2}\left ( x+5 \right ) = \log_{2} 6$

$\log_{2} \left[ \left ( x+4 \right )\left ( x+5 \right ) \right ] = \log_{2} 6$

$2^{ \log_{2} \left[ \left ( x+4 \right )\left ( x+5 \right ) \right ] }=2^{ \log_{2} 6}$

(x + 4)(x + 5) = 6

FOIL: $x^{2} + 4x + 5x + 4 \cdot 5 = 6$

$x^{2} + 9x + 20 = 6$

$x^{2} + 9x + 20 - 6 = 6- 6$

$x^{2} + 9x + 14 = 0$

(x + 2) (x + 7) = 0

$x + 2 = 0 \textrm{ or }x + 7 = 0$

$x = -2 \textrm{ or } x = -7$

These are our possible solutions. However, we need to test them.

$\log_{2}\left ( x+4 \right ) + \log_{2}\left ( x+5 \right ) = \log_{2} 6$

$\log_{2}\left ( -2+4 \right ) + \log_{2}\left ( -2+5 \right ) = \log_{2} 6$

$\log_{2}2 + \log_{2}3 = \log_{2} 6$

$\log_{2}2 = 1$

$\log_{2}3 = \frac{\ln 3}{\ln 2} \approx \frac{1.10}{0.69} \approx 1.59$

$\log_{2}6 = \frac{\ln 6}{\ln 2} \approx \frac{1.79}{0.69} \approx 2.59$

The equation becomes 1 + 1.59 = 2.59 . This is true, so x = -2 is a solution.

x = -7 :

$\log_{2}\left ( x+4 \right ) + \log_{2}\left ( x+5 \right ) = \log_{2} 6$

$\log_{2}\left ( -7+4 \right ) + \log_{2}\left ( -7+5 \right ) = \log_{2} 6$

$\log_{2}\left (-3 \right ) + \log_{2}\left (-2 \right ) = \log_{2} 6$

However, negative numbers do not have logarithms, so this equation is meaningless. x = -7 is not a solution, and x = -2 is the one and only solution. Since this is not one of our choices, the correct response is "The correct solution set is not included among the other choices."

Report an Error

Example Question #1 : Simplifying Exponential Functions

$\textup{Which of the following expressions is equivalent to }\sqrt{\frac{x^{4}y^{11}z^{5}}{x^{2}y^{5}z^{7}}} \textup{ ?}$

Possible Answers:

$\frac{xy^{4}}{z}$

$\frac{xy^{3}}{z^{2}}$

$\textup{None of the other answers}$

$\frac{xy^{3}}{z}$

$\frac{x^{2}y^{3}}{x^{2}}$

Correct answer:

$\frac{xy^{3}}{z}$

Explanation:

$\textup{Reduce the fraction by subtracting exponents, then halve all exponents to find the square root.}$

$\sqrt{\frac{x^{4}y^{11}z^{5}}{x^{2}y^{5}z^{7}}} =\sqrt{\frac{x^{2}y^{6}}{z^{2}}}\:\:=\frac{xy^{3}}{z}$

Report an Error

Example Question #1 : Sequences And Series

Evaluate: $1 - \frac{2}{3} + \frac{4}{9} - \frac{8}{27} + ...$

Possible Answers:

None of the other answers are correct.

$\frac{3}{5}$

$\frac{2}{5}$

$\frac{5}{3}$

Correct answer:

$\frac{3}{5}$

Explanation:

This sum can be determined using the formula for the sum of an infinite geometric series, with initial term $a_{0} = 1$ and common ratio $r = -\frac{2}{3}$ :

$S = \frac{a_{0}} {1-r} = \frac{1} {1- \left ( -\frac{2}{3}\right ) } = \frac{1} {1+ \frac{2}{3} }= \frac{1} { \frac{5}{3} } =\frac{3}{5}$

Report an Error

Example Question #1 : Sequences And Series

The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?

Possible Answers:

220

105

210

110

Correct answer:

220

Explanation:

An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it.

Let $a_{n}$ denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general:

a_n=a_1 +(n-1)d , where d is the common difference between two consecutive terms.

We are given the 4th and 8th terms in the sequence, so we can write the following equations:

a_4=a_1+(4-1)d=a_1+3d=-20

a_8=a_1+(8-1)d=a_1+7d=-10 .

We now have a system of two equations with two unknowns:

a_1+3d=-20

a_1+7d=-10

Let us solve this system by subtracting the equation a_1+7d=-10 from the equation a_1+3d=-20 . The result of this subtraction is

-4d=-10 .

This means that d = 2.5.

Using the equation a_1+7d=-10 , we can find the first term of the sequence.

a_1+7(2.5)=-10

a_1=-27.5

Ultimately, we are asked to find the hundredth term of the sequence.

$a_{100}=a_1+(100-1)d=-27.5+99(2.5)=220$

The answer is 220.

Report an Error

Example Question #3 : Sequences And Series

Find the sum, if possible:

$3+1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...$

Possible Answers:

$\frac{3}{2}$

$\frac{9}{2}$

Correct answer:

$\frac{9}{2}$

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms. The key here is finding the rate, or pattern, between the terms. Because this is a geometric sequence, the rate is the constant by which each new term is multiplied.

Plugging in our values, we get:

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

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

$S = \frac{9}{2}$

Report an Error

Example Question #1 : Sequences And Series

Find the sum, if possible:

$8-6+\frac{9}{2}-\frac{27}{8}+...$

Possible Answers:

$\frac{32}{7}$

$\frac{30}{7}$

$\frac{26}{7}$

$\frac{28}{7}$

$\frac{34}{7}$

Correct answer:

$\frac{32}{7}$

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms in a series

Because the terms switch sign, we know that the rate must be negative.

Plugging in our values, we get:

$S = \frac{8}{1-(\frac{-3}{4})}$

$S = \frac{8}{\frac{7}{4}}$

$S = \frac{32}{7}$

Report an Error

← Previous 1 2 3 4 5 6 Next →

View Tutors

Sumit
Certified Tutor

Calcutta University, Bachelor of Science, Mathematics. Indian Statistical Institute, Master of Science, Mathematics.

View Tutors

Cheryl
Certified Tutor

Drew University, Bachelor in Arts, Psychology. Kean University, Master of Arts, Educational Psychology.

View Tutors

Yucheng
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Mechanical Engineering.

All High School Math Resources

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

Popular Subjects

GRE Tutors in Washington DC, Biology Tutors in San Francisco-Bay Area, GMAT Tutors in Boston, Reading Tutors in Chicago, Computer Science Tutors in Phoenix, LSAT Tutors in Atlanta, Calculus Tutors in Miami, Physics Tutors in Phoenix, SSAT Tutors in Philadelphia, Math Tutors in Chicago

Popular Courses & Classes

SAT Courses & Classes in San Diego, GRE Courses & Classes in San Diego, Spanish Courses & Classes in Chicago, GRE Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Boston, MCAT Courses & Classes in Los Angeles, ACT Courses & Classes in Phoenix, SAT Courses & Classes in Boston, MCAT Courses & Classes in New York City, GRE Courses & Classes in Washington DC

Popular Test Prep

SSAT Test Prep in San Diego, SSAT Test Prep in Atlanta, LSAT Test Prep in Seattle, ISEE Test Prep in Houston, SSAT Test Prep in Miami, MCAT Test Prep in San Diego, SAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Houston, MCAT Test Prep in Seattle, SSAT Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

High School Math : Pre-Calculus

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #2 : Simplifying Polynomial Functions

Example Question #1 : Exponential And Logarithmic Functions

Example Question #21 : Pre Calculus

Example Question #2 : Exponential And Logarithmic Functions

Example Question #21 : Pre Calculus

Example Question #1 : Simplifying Exponential Functions

Example Question #1 : Sequences And Series

Example Question #1 : Sequences And Series

Example Question #3 : Sequences And Series

Example Question #1 : Sequences And Series

All High School Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: