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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #4693 : Algebra Ii

What is the value of $i^{42}$ $\displaystyle i^{42}$, if $\small i$ $\displaystyle \small i$= $\small \sqrt{-1}$ $\displaystyle \small \sqrt{-1}$ ?

Possible Answers:

$\displaystyle 1$

$\displaystyle -1$

$\textup{No solution}$ $\displaystyle \textup{No solution}$

$\small -i$ $\displaystyle \small -i$

$\small i$ $\displaystyle \small i$

Correct answer:

$\displaystyle -1$

Explanation:

We know that $\small i=\sqrt{-1}}$ $\displaystyle \small i=\sqrt{-1}}$. Therefore, $\small i^{2}=-1, i^{3}=-\sqrt{-1}, i^{4}=1$ $\displaystyle \small i^{2}=-1, i^{3}=-\sqrt{-1}, i^{4}=1$. Thus, every exponent of $\small i$ $\displaystyle \small i$ that is a multiple of 4 will yield the value of $\small 1$ $\displaystyle \small 1$. This makes $\small i^{40} =1$ $\displaystyle \small i^{40} =1$. Since $\small i^{42}=i^{40}*i*i$ $\displaystyle \small i^{42}=i^{40}*i*i$, we know that $\small i^{42}=1*i*i=i^{2}=-1$ $\displaystyle \small i^{42}=1*i*i=i^{2}=-1$.

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Example Question #3 : Complex Numbers

$\displaystyle (3+2i)+4-i-i(7+i)$

Possible Answers:

The answer is not present.

7-6i $\displaystyle 7-6i$

7(1-i) $\displaystyle 7(1-i)$

$\displaystyle -i$

8-6i $\displaystyle 8-6i$

Correct answer:

8-6i $\displaystyle 8-6i$

Explanation:

$\displaystyle (3+2i)+4-i-i(7+i)$

Combine like terms:

$\displaystyle 7+i-i(7+i)$

Distribute:

$\displaystyle 7+i-7i-i^2$

Combine like terms:

$\displaystyle 7+i-7i-(-1)$

$\mathbf{8-6i}$ $\displaystyle \mathbf{8-6i}$

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Example Question #4692 : Algebra Ii

$\sqrt{-4}*\sqrt{-25}*\sqrt{-64}$ $\displaystyle \sqrt{-4}*\sqrt{-25}*\sqrt{-64}$

Possible Answers:

-80i $\displaystyle -80i$

80i $\displaystyle 80i$

-8i $\displaystyle -8i$

$\displaystyle 8i$

80i^3 $\displaystyle 80i^3$

Correct answer:

-80i $\displaystyle -80i$

Explanation:

$\sqrt{-4}*\sqrt{-25}*\sqrt{-64}$ $\displaystyle \sqrt{-4}*\sqrt{-25}*\sqrt{-64}$

$\displaystyle 2i*5i*8i=80i^3$

i^3=-1 $\displaystyle i^3=-1$

$\mathbf{-80i}$ $\displaystyle \mathbf{-80i}$

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Example Question #2 : Complex Numbers

Rationalize the complex fraction: $\frac{2+i}{3-i}$ $\displaystyle \frac{2+i}{3-i}$

Possible Answers:

$\frac{5+5i}{8}$ $\displaystyle \frac{5+5i}{8}$

$\frac{7+5i}{10}$ $\displaystyle \frac{7+5i}{10}$

$\frac{5+i}{10}$ $\displaystyle \frac{5+i}{10}$

$\frac{1-i}{2}$ $\displaystyle \frac{1-i}{2}$

$\frac{1+i}{2}$ $\displaystyle \frac{1+i}{2}$

Correct answer:

$\frac{1+i}{2}$ $\displaystyle \frac{1+i}{2}$

Explanation:

To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.

$\frac{(2+i)(3+i)}{(3-i)(3+i)}$ $\displaystyle \frac{(2+i)(3+i)}{(3-i)(3+i)}$

$\frac{6+3i+2i+i^2}{9-3i+3i-i^2}$ $\displaystyle \frac{6+3i+2i+i^2}{9-3i+3i-i^2}$

$\frac{5+5i}{10}$ $\displaystyle \frac{5+5i}{10}$

$\frac{1+i}{2}$ $\displaystyle \frac{1+i}{2}$

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Example Question #2 : Complex Numbers

Rationalize the complex fraction: $\frac{2+i}{3-i}$ $\displaystyle \frac{2+i}{3-i}$

Possible Answers:

$\frac{5+5i}{8}$ $\displaystyle \frac{5+5i}{8}$

$\frac{7+5i}{10}$ $\displaystyle \frac{7+5i}{10}$

$\frac{5+i}{10}$ $\displaystyle \frac{5+i}{10}$

$\frac{1-i}{2}$ $\displaystyle \frac{1-i}{2}$

$\frac{1+i}{2}$ $\displaystyle \frac{1+i}{2}$

Correct answer:

$\frac{1+i}{2}$ $\displaystyle \frac{1+i}{2}$

Explanation:

To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.

$\frac{(2+i)(3+i)}{(3-i)(3+i)}$ $\displaystyle \frac{(2+i)(3+i)}{(3-i)(3+i)}$

$\frac{6+3i+2i+i^2}{9-3i+3i-i^2}$ $\displaystyle \frac{6+3i+2i+i^2}{9-3i+3i-i^2}$

$\frac{5+5i}{10}$ $\displaystyle \frac{5+5i}{10}$

$\frac{1+i}{2}$ $\displaystyle \frac{1+i}{2}$

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Example Question #2031 : Mathematical Relationships And Basic Graphs

Multiply: $\displaystyle (3i + 1 )(3 - 4i)$

Possible Answers:

2 - 12i $\displaystyle 2 - 12i$

5i-9 $\displaystyle 5i-9$

5i+15 $\displaystyle 5i+15$

3-7i $\displaystyle 3-7i$

Correct answer:

5i+15 $\displaystyle 5i+15$

Explanation:

Distribute:

$\displaystyle 3i(3-4i) = 9i + 12$

$\displaystyle 1(3-4i) = 3 - 4i$

combine like terms:

5i + 15 $\displaystyle 5i + 15$

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Example Question #4 : Complex Numbers

Multiply: $\displaystyle (3+i)(5-4i)$

Possible Answers:

15-11i $\displaystyle 15-11i$

11-7i $\displaystyle 11-7i$

15-3i $\displaystyle 15-3i$

19-7i $\displaystyle 19-7i$

19+17i $\displaystyle 19+17i$

Correct answer:

19-7i $\displaystyle 19-7i$

Explanation:

Use FOIL to multiply the two binomials.

Recall that FOIL stands for Firsts, Outers, Inners, and Lasts.

$\displaystyle (3+i)(5-4i)$

$\displaystyle 15+5i-12i-4i^2$

Remember that i^2=-1 $\displaystyle i^2=-1$

15-7i+4 $\displaystyle 15-7i+4$

19-7i $\displaystyle 19-7i$

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Example Question #2032 : Mathematical Relationships And Basic Graphs

Simplify: $\displaystyle (5-3i)-(2+2i)$

Possible Answers:

3-5i $\displaystyle 3-5i$

7-i $\displaystyle 7-i$

7+i $\displaystyle 7+i$

3-i $\displaystyle 3-i$

$\displaystyle 8$

Correct answer:

3-5i $\displaystyle 3-5i$

Explanation:

Distribute the minus sign:

$\displaystyle 5-3i-2-2i$

Combine like terms:

3-5i $\displaystyle 3-5i$

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Example Question #21 : Basic Operations With Complex Numbers

Simplify:

$\displaystyle (3+i)(1-2i)$

Possible Answers:

5+5i $\displaystyle 5+5i$

1-2i $\displaystyle 1-2i$

1-5i $\displaystyle 1-5i$

5-5i $\displaystyle 5-5i$

3-2i $\displaystyle 3-2i$

Correct answer:

5-5i $\displaystyle 5-5i$

Explanation:

Use FOIL to multiply the binomials:

$\displaystyle 3-6i+i-2i^2$

Change i^2 $\displaystyle i^2$ to -1:

$\displaystyle 3-6i+i+2$

Combine like terms:

5-5i $\displaystyle 5-5i$

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Example Question #84 : Imaginary Numbers

Find $i^{123}$ $\displaystyle i^{123}$

Possible Answers:

$\displaystyle 1$

$\displaystyle -1$

$\displaystyle i$

$\displaystyle -i$

$\displaystyle 0$

Correct answer:

$\displaystyle -i$

Explanation:

i^1=i $\displaystyle i^1=i$

i^2=-1 $\displaystyle i^2=-1$

i^3=-i $\displaystyle i^3=-i$

i^4=1 $\displaystyle i^4=1$

This pattern repeats every four powers. Divide the power by 4:

$\frac{123}{4}=30 R3$ $\displaystyle \frac{123}{4}=30 R3$

Since the remainder is 3, it is equivalent to i^3 $\displaystyle i^3$, or $\displaystyle -i$.

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #4693 : Algebra Ii

Example Question #3 : Complex Numbers

Example Question #4692 : Algebra Ii

Example Question #2 : Complex Numbers

Example Question #2 : Complex Numbers

Example Question #2031 : Mathematical Relationships And Basic Graphs

Example Question #4 : Complex Numbers

Example Question #2032 : Mathematical Relationships And Basic Graphs

Example Question #21 : Basic Operations With Complex Numbers

Example Question #84 : Imaginary Numbers

All Algebra II Resources

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