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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #81 : Basic Operations With Complex Numbers

Select the complex conjugate of .

Possible Answers:

-6i

has no complex conjugate.

Correct answer:

-6i

Explanation:

can be restated in standard complex number form as

0+6i .

The complex conjugate of a complex number a+bi is a-bi , so the complex conjugate of 0+6i is 0-6i , which is equal to -6i . Therefore, -6i is the complex conjugate of .

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Example Question #82 : Basic Operations With Complex Numbers

Select the complex conjugate of .

Possible Answers:

-9i

9 has no complex conjugate.

Correct answer:

Explanation:

can be restated in standard complex number form as

-9 + 0i .

The complex conjugate of a complex number a+bi is a-bi , so the complex conjugate of -9 + 0i is -9 - 0i , which is also equal to . Therefore, itself is the complex conjugate of .

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Example Question #83 : Basic Operations With Complex Numbers

Select the complex conjugate of $2 + \frac{1}{3} i$ .

Possible Answers:

$\frac{1}{3} + 2i$

$-2 + \frac{1}{3} i$

$\frac{1}{3} - 2i$

$-2 - \frac{1}{3} i$

$2 - \frac{1}{3} i$

Correct answer:

$2 - \frac{1}{3} i$

Explanation:

The complex conjugate of a complex number a+bi is a-bi , so the complex conjugate of $2 + \frac{1}{3} i$ is $2 - \frac{1}{3} i$ .

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Example Question #84 : Basic Operations With Complex Numbers

Subtract $7 + i \sqrt{17}$ from its complex conjugate. What is the result?

Possible Answers:

$2 i\sqrt{17}$

$-2 i\sqrt{17}$

Correct answer:

$-2 i\sqrt{17}$

Explanation:

The complex conjugate of a complex number a+bi is a-bi .

Therefore, the complex conjugate of $7 + i \sqrt{17}$ is $7 - i \sqrt{17}$ . Subtract the former from the latter:

$\left (7 - i \sqrt{17} \right )-\left ( 7 + i \sqrt{17} \right )$

Collect real parts and imaginary parts:

$= 7 - 7 - i \sqrt{17} - i \sqrt{17}$

The real parts cancel out:

$= - i \sqrt{17} - i \sqrt{17}$

$= -2 i \sqrt{17}$

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Example Question #141 : Imaginary Numbers

Evaluate:

$i^{-47}$

Possible Answers:

Correct answer:

Explanation:

In order to raise to the power of a negative integer, first raise to the absolute value of that integer. Therefore, to find $i^{-47}$ , we need to look at $i^{47}$ .

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the following table:

Powers of i

$47 \div 4 = 11 \textrm{ R }3$ , so $i^{47}$ can be determined by selecting the power of corresponding to remainder 3. This is .

Since $i^{47} = -i$ , and by definition, $a^{-N} = \frac{1}{a^{N}}$ , it follows that

$i^{-47} = \frac{1}{i^{47}} = \frac{1}{-i}$

Rationalize the denominator by multiplying by , the complex conjugate:

$i^{-47} = \frac{1}{-i} = \frac{1 \cdot i}{-i \cdot i} = \frac{i}{-i^{1}} = \frac{i}{-(-1)} = \frac{i}{1} = i$

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Example Question #86 : Basic Operations With Complex Numbers

Evaluate:

$i^{64}$

Possible Answers:

Correct answer:

Explanation:

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Powers of i

$64 \div 4 = 16\textrm{ R }0$ , so $i^{64}$ can be determined by selecting the power of corresponding to remainder 0. The corresponding power is 1, so $i^{64} = 1$ .

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Example Question #151 : Imaginary Numbers

Evaluate:

$i^{73 }$

Possible Answers:

Correct answer:

Explanation:

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Powers of i

$73 \div 4 = 18 \textrm{ R }1$ , so $i^{73 }$ can be determined by selecting the power of corresponding to remainder 1. The correct power is , so $i^{73 } =i$ .

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Example Question #152 : Imaginary Numbers

Evaluate:

$| 9- i\sqrt{6}|$

Possible Answers:

None of these

$9- \sqrt{6}$

$9+ \sqrt{6}$

$\sqrt{87}$

$5 \sqrt{3}$

Correct answer:

$\sqrt{87}$

Explanation:

|a-bi| refers to the absolute value of a complex number a-bi , which can be calculated by evaluating $\sqrt{a^{2}+ b^{2}}$ . Setting $a= 9, b = \sqrt{6}$ , the value of this expression is

$| 9- i\sqrt{6}| = \sqrt{9^{2}+ (\sqrt{6})^{2}} = \sqrt{81+ 6} = \sqrt{87}$

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Example Question #91 : Basic Operations With Complex Numbers

Evaluate:

| 5 + 12 i |

Possible Answers:

Correct answer:

Explanation:

|a+bi| refers to the absolute value of a complex number a+bi , which can be calculated by evaluating $\sqrt{a^{2}+ b^{2}}$ . Setting a= 5, b = 12 , the value of this expression is:

$| 5 + 12 i | = \sqrt{5^{2}+ 12^{2}} = \sqrt{25+ 144} = \sqrt{169} = 13$

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Example Question #1 : Equations With Complex Numbers

x, y are real numbers.

$4x+ \left (3y+ 4 \right ) i = 21 + 7i$

Evaluate .

Possible Answers:

y = 1

y = 6.25

y = -1

y =- 5.25

y = 5.25

Correct answer:

y = 1

Explanation:

For two imaginary numbers to be equal to each other, their imaginary parts must be equal. Therefore, we set, and solve for in:

3y + 4 = 7

3y = 3

y = 1

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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 #81 : Basic Operations With Complex Numbers

Example Question #82 : Basic Operations With Complex Numbers

Example Question #83 : Basic Operations With Complex Numbers

Example Question #84 : Basic Operations With Complex Numbers

Example Question #141 : Imaginary Numbers

Example Question #86 : Basic Operations With Complex Numbers

Example Question #151 : Imaginary Numbers

Example Question #152 : Imaginary Numbers

Example Question #91 : Basic Operations With Complex Numbers

Example Question #1 : Equations With Complex Numbers

All Algebra II Resources

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