Algebra II : Complex Imaginary Numbers

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #21 : Imaginary Numbers

Simplify:

Possible Answers:

None of the above

Correct answer:

Explanation:

To solve a radical that has a negative sign under it we need to factor it first.

Recall that . Using this fact we get the following.

 

Example Question #21 : Complex Imaginary Numbers

Simplify:

Possible Answers:

None of the above

Correct answer:

Explanation:

To simplify a radical that has a negative sign under it we need to factor it first.

After factoring it recall that .

Therefore we get the following.

Example Question #23 : Imaginary Numbers

Simplify:

Possible Answers:

Correct answer:

Explanation:

To simplify, we must FOIL:

Remembering that , we can simplify this further to

Example Question #24 : Imaginary Numbers

Simplify:  

Possible Answers:

Correct answer:

Explanation:

Rewrite the expression and eliminate the negative exponent.

Evaluate .

We can rewrite  as an exponential product using .

Rewrite the expression.

The answer is:  

Example Question #25 : Imaginary Numbers

Possible Answers:

Correct answer:

Explanation:

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The number 1680 divides evenly by 4. This means that 1680 completes the cycle and has the same answer as

Example Question #21 : Complex Imaginary Numbers

Simplify

Possible Answers:

Correct answer:

Explanation:

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The closest number under 358 that evenly divides by 4 is 356. 356 completes the cycle. 358 is two numbers into the start of a new cycle and has the same value as

Example Question #27 : Imaginary Numbers

Simplify

Possible Answers:

Correct answer:

Explanation:

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The closest number under 273 that divides by 4 is 272. This means the next number 273 will be the start of a new cycle and have the same answer as

Example Question #28 : Imaginary Numbers

Simplify

Possible Answers:

Correct answer:

Explanation:

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The closest number under the exponent that divides evenly by 4 is 700. This means that 703 is the third number in a new cycle and will have the same value as

Example Question #29 : Imaginary Numbers

Solve:

Possible Answers:

None of these

Correct answer:

Explanation:

Definition of

Thus,

Example Question #30 : Imaginary Numbers

Simplify the expression:  

Possible Answers:

Correct answer:

Explanation:

Multiply the numerator and denominator by the conjugate of the denominator.

Multiply both the top and bottom by using the FOIL method.

Numerator:  

Recall that , which means that .

Denominator:

Divide the numerator with the denominator.

The answer is:  

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