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High School Math : Pre-Calculus

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Example Questions

Example Question #1 : Simplifying Polynomial Functions

Simplify the following polynomial function:

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

Possible Answers:

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

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

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

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

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

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:

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

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Example Question #1 : Solving Logarithms

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:

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

$\frac{2x}{y}$

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

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

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

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 $10^{B} = y$

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

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Example Question #21 : Pre Calculus

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

Possible Answers:

Correct answer:

Explanation:

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

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

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Example Question #2 : Solving Logarithms

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 .

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Example Question #1 : Simplifying Logarithms

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

Possible Answers:

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

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

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

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

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."

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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:

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

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

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

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

$\textup{None of the other answers}$

Correct answer:

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

Explanation:

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

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

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Example Question #1 : Sequences And Series

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

Possible Answers:

$\frac{3}{5}$

$\frac{2}{5}$

None of the other answers are correct.

$\frac{5}{3}$

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 $r = -\frac{2}{3}$ :

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

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Example Question #1 : Arithmetic And Geometric 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

110

105

210

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.

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Example Question #1 : Sequences And Series

Find the sum, if possible:

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

Possible Answers:

$\frac{9}{2}$

$\frac{3}{2}$

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:

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

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

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

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Example Question #2 : Sequences And Series

Find the sum, if possible:

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

Possible Answers:

$\frac{34}{7}$

$\frac{30}{7}$

$\frac{26}{7}$

$\frac{32}{7}$

$\frac{28}{7}$

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:

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

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

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

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High School Math : Pre-Calculus

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Simplifying Polynomial Functions

Example Question #1 : Solving Logarithms

Example Question #21 : Pre Calculus

Example Question #2 : Solving Logarithms

Example Question #1 : Simplifying Logarithms

Example Question #1 : Simplifying Exponential Functions

Example Question #1 : Sequences And Series

Example Question #1 : Arithmetic And Geometric Series

Example Question #1 : Sequences And Series

Example Question #2 : Sequences And Series

All High School Math Resources

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