Understanding Imaginary and Complex Numbers

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Math › Understanding Imaginary and Complex Numbers

Questions 1 - 10

Multiply:

Explanation

Since and are conmplex conjugates, they can be multiplied according to the following pattern:

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

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

None of the other answer choices are correct.

Explanation

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

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

Evaluate: $i ^{45}$

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:

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

Which of the following is equivalent to:

$-4i^{4} ?$

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.

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

Explanation

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

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

What is the absolute value of

$\frac{3}{4}$

$\frac{-4}{3}$

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.

Simplify the expression.

None of the other answer choices are correct.

Explanation

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

Substitute to eliminate $\small i^2$ .

Simplify.

Simplify the radical.

$\sqrt{-250}$

$5i\sqrt{10}$

$25i\sqrt{10}$

No solution

Explanation

$\sqrt{-250}$

First, factor the term in the radical.

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

Now, we can simplify.

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

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

$5i\sqrt{10}$

Multiply:

Explanation

FOIL:

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

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

Multiply:

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

$51 + 2i\sqrt{2}$

$51 - 2i\sqrt{2}$

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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