Basic Operations with Complex Numbers

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Algebra 2 › Basic Operations with Complex Numbers

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CORRECT

Explanation

This problem requires you to use FOIL to multiply the binomials.

Multiply the first terms

$(5i\cdot i=5i^2)$ ,

then the outside terms

$(5i\cdot 3=15i)$ ,

next the inside terms

$(-2\cdot i=-2i)$ ,

and finally the last terms

$(-2\cdot 3=-6)$ .

Put those together to get

Recall that

Therefore, your answer is

$What\ is\ the\ complex\ conjugate\ for\ 9-2i?$

CORRECT

$81-4i^{2}$

$81+4i^{2}$

Explanation

$The\ complex\ conjugate\ for\ a binomial\ with\ an\ imaginary\ number\ is$

$simply\ the\ original\ with\ the\ sign\ for\ the\ imaginary\ number\ changed.$

$If\ 9-2i\ were\ multiplied\ by\ 9+2i\ the\ i\ values\ would\ cancel\ producing\ a\ real\ number.$

Solve: $-2i^2-4i^{-4}+6i^6$

CORRECT

Explanation

Evaluate each term of the expression. Write out the values of the imaginary terms.

$i=\sqrt{-1}$

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

$i^4 = i^2\cdot i^2 = (-1)\cdot (-1) = 1$

$i^6 = i^4 \cdot i^2 = (1)\cdot (-1) = -1$

Replace the values of each.

$-4i^{-4} = \frac{-4}{i^4} = -\frac{4}{1} = -4$

Sum all the values.

$-2i^2-4i^{-4}+6i^6 = 2-4-6 = -8$

The answer is:

Consider the following definitions of imaginary numbers:

Then,

CORRECT

Explanation

CORRECT

Explanation

Nothing can be simplified in either parentheses, so the first step is to distribute the negative sign to the second parentheses

Then, you combine similar terms remembering that terms with i cannot combine with those with no i

Evaluate:

CORRECT

Explanation

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

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

Compute: $-5i^{1234}$

CORRECT

Explanation

Identify the first two powers of the imaginary term.

$i=\sqrt{-1}$

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

Rewrite the expression as a product of exponents.

$-5i^{1234}=-5(i^{2})^{617} = -5(-1)^{617}$

Negative one to an odd power will be negative one.

The answer is:

Evaluate:

CORRECT

Explanation

Write the powers of the imaginary numbers.

$i=\sqrt{-1}$

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

$i^3 = i^2\cdot i = -i$

$i^4 = i^2\cdot i^2 = (-1)\cdot (-1) = 1$

Notice that this will repeat. We can rewrite higher powers if the imaginary term by product of powers.

The answer is:

Evaluate:

$i^{64}$

CORRECT

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 $i^{64} = 1$ .