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High School Math : Imaginary Numbers

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Example Questions

Example Question #1 : Imaginary Numbers

Simplify the radical.

$\sqrt{-250}$

Possible Answers:

$5i\sqrt{10}$

15i

No solution

$25i\sqrt{10}$

Correct answer:

$5i\sqrt{10}$

Explanation:

$\sqrt{-250}$

First, factor the term in the radical.

-250=(-1)(25)(10)

$\sqrt{-250}=(\sqrt{-1})(\sqrt{25})(\sqrt{10})$

Now, we can simplify.

$\sqrt{-1}=i\ \text{and}\ \sqrt{25}=5$

$(\sqrt{-1})(\sqrt{25})(\sqrt{10})=(i)(5)(\sqrt{10})$

$5i\sqrt{10}$

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Example Question #2 : Imaginary Numbers

Multiply: (3 + 2i)(4 - 3 i)

Possible Answers:

18-17i

18-i

6+ i

6 - 17i

6-i

Correct answer:

18-i

Explanation:

FOIL:

(3 + 2i)(4 - 3 i)

$= 3 \cdot 4 -3 \cdot 3i + 2i \cdot4 - 2i \cdot3i$

$= 12 -9i + 8i - 6i^{2}$

= 12 -9i + 8i - 6 (-1)

= 12 -9i + 8i +6

= 18-i

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Example Question #3 : Imaginary Numbers

Multiply: (5+4i) (5-4i)

Possible Answers:

25+16i

25-16i

Correct answer:

Explanation:

Since 5+4i and 5-4i are conmplex conjugates, they can be multiplied according to the following pattern:

$(5+4i) (5-4i) = 5 ^{2} + 4^{2} = 25 + 16 = 41$

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Example Question #4 : Imaginary Numbers

Multiply:

$\left (7 + i \sqrt{2 } \right )\left (7 - i \sqrt{2 } \right )$

Possible Answers:

$51 - 2i\sqrt{2}$

49-2i

$51 + 2i\sqrt{2}$

Correct answer:

Explanation:

Since $7 + i \sqrt{2 }$ and $7 - i \sqrt{2 }$ are conmplex conjugates, they can be multiplied according to the following pattern:

$\left (7 + i \sqrt{2 } \right )\left (7 - i \sqrt{2 } \right ) = 7^{2} +\left ( \sqrt{2} \right )^{2} = 49 + 2 = 51$

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Example Question #5 : Imaginary Numbers

Evaluate: $i ^{45}$

Possible Answers:

45i

Correct answer:

Explanation:

$i ^{N}$ can be evaluated by dividing by 4 and noting the remainder. Since $45 \div 4 = 11 \textrm{ R } 1$ - that is, since dividing 45 by 4 yields remainder 1:

$i ^{45} = i ^{1} = i$

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Example Question #6 : Imaginary Numbers

Evaluate: $(4 + 3i)^{2}$

Possible Answers:

7 + 24i

13 - 24i

19 + 24i

Correct answer:

7 + 24i

Explanation:

$(4 + 3i)^{2}$

$= 4^{2} + 2 \cdot 4 \cdot 3i +\left ( 3i \right )^{2}$

= 16 + 24i + 9 (-1)

= 16 + 24i -9

= 7 + 24i

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Example Question #1 : Basic Operations With Complex Numbers

What is the absolute value of 4-3i

Possible Answers:

$\frac{3}{4}$

$\frac{-4}{3}$

Correct answer:

Explanation:

The absolute value is a measure of the distance of a point from the origin. Using the pythagorean distance formula to calculate this distance.

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Example Question #1 : Imaginary Numbers

Which of the following is equivalent to:

$-4i^{4} ?$

Possible Answers:

-4i

Correct answer:

Explanation:

Recall that $i^{2} = -1$ .

Then, we have that $-4(i^{4}) = -4(i^{2})^{2} = -4(-1)^{2} = -4(1) = -4$ .

Note that we used the power rule of exponents and the order of operations to simplify the exponent before multiplying by the coefficient.

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Example Question #2 : Imaginary Numbers

Simplify the expression.

4i^2-6i-7i^2+3i+4

Possible Answers:

7-3i

-7-3i

-3i^2-3i+4

None of the other answer choices are correct.

Correct answer:

7-3i

Explanation:

Combine like terms. Treat $\small i$ as if it were any other variable.

4i^2-6i-7i^2+3i+4

-3i^2-3i+4

Substitute to eliminate $\small i^2$ .

i^2=-1

-3(-1)-3i+4

Simplify.

3-3i+4=7-3i

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Example Question #1 : Imaginary Numbers

Which of the following is equivalent to $5i^{2}$ ?

Possible Answers:

None of the other answer choices are correct.

Correct answer:

Explanation:

Recall the basic property of imaginary numbers, $i^{2} = -1$ .

Keeping this in mind, $5i^{2} = 5(-1) = -5$ .

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High School Math : Imaginary Numbers

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Imaginary Numbers

Example Question #2 : Imaginary Numbers

Example Question #3 : Imaginary Numbers

Example Question #4 : Imaginary Numbers

Example Question #5 : Imaginary Numbers

Example Question #6 : Imaginary Numbers

Example Question #1 : Basic Operations With Complex Numbers

Example Question #1 : Imaginary Numbers

Example Question #2 : Imaginary Numbers

Example Question #1 : Imaginary Numbers

All High School Math Resources

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