Basic Operations with Complex Numbers

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Algebra II › Basic Operations with Complex Numbers

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Evaluate:

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

$\sqrt{87}$

CORRECT

$5 \sqrt{3}$

$9+ \sqrt{6}$

$9- \sqrt{6}$

None of these

Explanation

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

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

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 $i^{64} = 1$ .

Evaluate:

CORRECT

Explanation

Rewrite the problem as separate groups of binomials.

Use the FOIL method to expand the first two terms.

Simplify the right side.

Recall that since $i=\sqrt{-1}$ , the value of $i^2 = i\cdot i =-1$ .

Multiply this value with the third binomial.

Simplify the terms.

The answer is:

Evaluate:

$i^{73 }$

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

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