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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #3 : Other Exponent Applications

Lucia deposits $40,000 into a savings account that pays 5.5% annual interest compounded continuously. Assuming she neither deposits nor withdraws money, how long will it take for her to have $60,000 in the account?

Possible Answers:

Between 6 and 7 years

Between 9 and 10 years

Between 7 and 8 years

Between 8 and 9 years

Between 5 and 6 years

Correct answer:

Between 7 and 8 years

Explanation:

We set final principal A = 60,000 original principal $A_{0} = 40,000$ , and interest rate r = 0.055 . We solve for in the continuous compound interest formula:

$A = A_{0} e ^{rt}$

$60,000 =40,000 e ^{0.055t}$

$\frac{60,000 }{40,000 }=\frac{40,000 e ^{0.055t}}{40,000 }$

$1.5 = e ^{0.055t}$

$0.055t = \ln 1.5$

$t = \frac{\ln 1.5}{0.055} \approx \frac{0.4055}{0.055} \approx 7.37$

The correct response is therefore between 7 and 8 years.

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Example Question #2 : Other Exponent Applications

Ann deposits $30,000 into a savings account that pays 4.5% annual interest compounded quarterly. Assuming she neither deposits nor withdraws money, what is the amount of time it will take for her to have at least $50,000 in the account?

Possible Answers:

$11 \textup{ yrs } 6\textup{ mo}$

$13\textup{ yrs }$

$11 \textup{ yrs }$

$12\textup{ yrs }$

$12 \textup{ yrs } 6\textup{ mo}$

Correct answer:

$11 \textup{ yrs } 6\textup{ mo}$

Explanation:

Apply the compound interest formula:

$A = A_{0} \left ( 1 + \frac{r}{n}\right ) ^{nt}$ .

We set final principal A = 50,000 original principal $A_{0} = 30,000$ , interest rate r = 0.045 , number of periods per year n = 4 (quarterly). We solve for in the equation

$50,000=30,000 \left ( 1 + \frac{0.045}{4}\right ) ^{4t}$

$50,000=30,000 \left ( 1 .01125 \right ) ^{4t}$

$\frac{50,000}{30,000}=\frac{30,000 \left ( 1 .01125 \right ) ^{4t}}{30,000}$

$1.6667= \left ( 1 .01125 \right ) ^{4t}$

$\log 1.6667= \log \left ( 1 .01125 \right ) ^{4t}$

$\log 1.6667= 4t \log 1 .01125$

$t = \frac{\log 1.6667}{4\log 1 .01125} \approx \frac{0.2219}{4 \cdot 0.004859} \approx 11.4$

This is rounded up to the next quarter of a year, so the correct response is 11.5 years, or 11 years 6 months.

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Example Question #801 : Mathematical Relationships And Basic Graphs

A company is constructing a wall with 4-sides, all sides are of equal length.

Write an equation using exponents to calculate the area of the wall. Use as the length and height.

Possible Answers:

$A=y\cdot y \cdot y \cdot y$

A=y^4

A=y+y

A=y^2

$A=y\cdot y$

Correct answer:

A=y^2

Explanation:

The formula to find area is:

$= length\cdot width=area$ is correct.

In our case our length is equal to our width which is .

Substituting our values into our equation we get:

$y\cdot y= A$

y^2=A

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Example Question #1 : Intermediate Single Variable Algebra

Simplify.

$x^{-6}x^3x^9$

Possible Answers:

$x^{-12}$

x^6

$x^{12}$

$x^{-6}$

Correct answer:

x^6

Explanation:

Put the negative exponent on the bottom so that you have $\frac{x^{12}}{x^{6}}$ which simplifies further to $x^{6}$ .

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Example Question #3461 : Algebra Ii

Evaluate the following expression

$(5^{0})^{2}$

Possible Answers:

$\frac{1}{25}$

$\frac{1}{2}$

$5^{20}$

Correct answer:

Explanation:

$(5^{0})^{2}=(1)^{2}=1$

We can also solve this problem using a different apporach

$(5^{0})^{2}=5^{0\times2}=5^{0}=1$

Remember that any number raised to the 0th power equals 1

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Example Question #2 : Simplifying Exponents

Simplify the following expression

$((yz)^2)^{3}$

Possible Answers:

$y^{2}z^{2}$

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

$y^{6}z^{6}$

Correct answer:

$y^{6}z^{6}$

Explanation:

$((yz)^{2})^{3}=(yz)^{6}=y^{6}z^{6}$

Alternatively,

$((yz)^{2})^{3}=(y^{2}z^{2})^3=y^{2\times3}z^{2\times3}=y^{6}z^{6}$

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Example Question #1 : Multiplying And Dividing Exponents

Simplify the following expression

$((6^2)^{0})^{3}$

Possible Answers:

36^3

6^6

6^5

Correct answer:

Explanation:

$((6^2)^{0})^{3}=6^{2\times 0\times 3}=6^0=1$

Remember that any number raised to the 0th power equals 1

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Example Question #4 : Simplifying Exponents

Evaluate the following expression

$\left (\frac{10^{3}}{10^{0}} \right )^{2}$

Possible Answers:

1,000,000

100,000

Correct answer:

1,000,000

Explanation:

$\left (\frac{10^{3}}{10^{0}} \right )^{2}=\frac{10^{3\times2}}{10^{0\times2}}=\frac{10^{6}}{10^{0}}=\frac{1000000}{1}=1,000,000$

Alternatively,

$\left (\frac{10^{3}}{10^{0}} \right )^{2}=\left (10^{3-0} \right )^{2}=\left (10^{3} \right )^{2}=10^{3\times 2}=10^{6}=1,000,000$

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Example Question #1 : Simplifying Exponents

Simplify:

$\frac{4a^{-5}b^{3}c^{2}}{12a^{-7}b^{-5}c^{-2}}$

Possible Answers:

$\frac{4a^{2}b^{2}}{12}$

$\frac{a^{-12}b^{-2}c^{0}}{3}$

$\frac{1}{3a^{-12}b^{-2}}$

$3a^{12}b^{-8}$

$\frac{a^{2}b^{8}c^{4}}{3}$

Correct answer:

$\frac{a^{2}b^{8}c^{4}}{3}$

Explanation:

$\frac{4a^{-5}b^{3}c^{2}}{12a^{-7}b^{-5}c^{-2}}$

Step 1: Simplify the exponents using the division of exponents rule (subtract exponents in demoninator from exponents in numerator).

$\frac{4a^{-5-(-7)}b^{3-(-5)}c^{2-(-2)}}{12}$

$\frac{4a^{2}b^{8}c^{4}}{12}$

Step 2: Reduce the fraction

$\frac{a^{2}b^{8}c^{4}}{3}$

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Example Question #1 : Simplifying Exponents

Simplify:

$\frac{q^{7}r^{5}s^{16}}{q^{2}r^{3}s}$

Possible Answers:

$q^{9}r^{8}s^{17}$

$q^{14}r^{15}s^{16}$

qrs

$q^{5}r^{2}s^{15}$

$q^{3.5}r^{1.3}s^{16}$

Correct answer:

$q^{5}r^{2}s^{15}$

Explanation:

Follow the division of exponents rule. Subract the exponents in the denominator from the exponents in the numerator.

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3 : Other Exponent Applications

Example Question #2 : Other Exponent Applications

Example Question #801 : Mathematical Relationships And Basic Graphs

Example Question #1 : Intermediate Single Variable Algebra

Example Question #3461 : Algebra Ii

Example Question #2 : Simplifying Exponents

Example Question #1 : Multiplying And Dividing Exponents

Example Question #4 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

All Algebra II Resources

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