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Algebra II : Imaginary Numbers

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #2071 : Mathematical Relationships And Basic Graphs

Evaluate: $2i^2-100i^{100}$

Possible Answers:

-102-i

-2-i

-2i

-100+i

Correct answer:

Explanation:

Write the first few terms of the imaginary powers.

$i=\sqrt{-1}$

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

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

The term with $i^{100}$ can be rewritten as a product of exponents.

$i^{100} = (i^4)^{25} = (1)^{25}=1$

Re-substitute the values.

$2i^2-100i^{100}= 2(-1)-100(1) = -2-100 = -102$

The answer is:

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Example Question #2071 : Mathematical Relationships And Basic Graphs

Evaluate: (2i^4+3)(-i^2-6)(i^6-i^4)

Possible Answers:

$\textup{Cannot be determined.}$

100

Correct answer:

Explanation:

Do not use the FOIL method to simplify these terms. Instead, write out the first few terms of referring to the problem, as well as the value of i^6 .

$i=\sqrt{-1}$

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

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

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

Substitute all the values of the imaginary numbers to the expression.

(2(1)+3)(-(-1)-6)((-1)-(1))

Simplify the terms.

(5)(-5)(-2) = 50

The answer is:

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Example Question #61 : Basic Operations With Complex Numbers

Add: 2i^2+9i^4-3i^8

Possible Answers:

Correct answer:

Explanation:

Write the imaginary values of the powers given. Recall that $i=\sqrt{-1}$ .

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

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

$i^8 = i^4\cdot i^4 = 1\cdot 1 =1$

Substitute the values into the expression.

2i^2+9i^4-3i^8 = 2(-1)+9(1)-3(1) = -2+9-3

The answer is:

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Example Question #121 : Imaginary Numbers

Evaluate: $-(i^{-36})$

Possible Answers:

2-i

-2i

Correct answer:

Explanation:

Rewrite the expression without the negative exponent.

$x^{-a} = \frac{1}{x^a}$

$-(i^{-36}) = -(\frac{1}{i^{36}})=-\frac{1}{i^{36}}$

Write out the first few terms of the imaginary number.

$i=\sqrt{-1}$

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

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

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

This will repeat for higher powers.

The power of 36 goes into 4 nine times. We can rewrite the exponent as a product of exponents.

$-\frac{1}{i^{36}} = -\frac{1}{(i^4)^9}$

Replace i^4 with the known value.

$-\frac{1}{(i^4)^9} = -\frac{1}{(1)^9} = -1$

The answer is:

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Example Question #63 : Basic Operations With Complex Numbers

Solve: $[(i^2)^3]^{-4}$

Possible Answers:

$\textup{The answer is not given.}$

Correct answer:

Explanation:

Write the values of the first two imaginary terms.

$i=\sqrt{-1}$

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

Replace the term inside the expression.

$[(-1)^3]^{-4} = [-1]^{-4}$

Rewrite the negative exponent into a fraction.

$\frac{1}{(-1)^4} = \frac{1}{1} = 1$

The answer is:

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Example Question #61 : Basic Operations With Complex Numbers

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

Possible Answers:

Correct answer:

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.

-2i^2 = -2(-1) = 2

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

6i^6 = 6(-1) =-6

Sum all the values.

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

The answer is:

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Example Question #65 : Basic Operations With Complex Numbers

Solve: -i^2-2i^4-15i^6

Possible Answers:

Correct answer:

Explanation:

Write out the terms of the imaginary powers.

$i=\sqrt{-1}$

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

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

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

Replace all terms with the correct value.

-i^2-2i^4-15i^6=-(-1)-2(1)-15(-1)

1-2+15= 14

The answer is:

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Example Question #66 : Basic Operations With Complex Numbers

Add the following terms: $i^6-i^4-3i^2-i^{10}$

Possible Answers:

Correct answer:

Explanation:

Simplify each term. Recall that:

$i=\sqrt{-1}$

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

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

This means that:

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

$i^{10} =( i^2)^5 = (-1)^{5} = -1$

Subtract the terms.

$i^6-i^4-3i^2-i^{10}=-1-1-3(-1)-(-1)$

Simplify the terms.

-2+3+1 = 2

The answer is:

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Example Question #67 : Basic Operations With Complex Numbers

Solve: $\frac{2i^3-9i^5}{i}$

Possible Answers:

9i+2

-9i+2

Correct answer:

Explanation:

Simplify by dividing the imaginary term.

$\frac{2i^3-9i^5}{i} = 2i^2-9i^4$

Write the first few terms of the imaginary powers.

$i=\sqrt{-1}$

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

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

Substitute the values into the expression.

2i^2-9i^4 = 2(-1)-9(1) = -2-9 = -11

The answer is:

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Example Question #121 : Imaginary Numbers

(6-3i)-(3+6i)

Possible Answers:

9-3i

None of these.

3+9i

-9i-3

3-9i

Correct answer:

3-9i

Explanation:

(6-3i)-(3+6i)

Distribute the negative sign:

6-3i-3-6i

Combine like terms:

$\mathbf{3-9i}$

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Algebra II : Imaginary Numbers

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #2071 : Mathematical Relationships And Basic Graphs

Example Question #2071 : Mathematical Relationships And Basic Graphs

Example Question #61 : Basic Operations With Complex Numbers

Example Question #121 : Imaginary Numbers

Example Question #63 : Basic Operations With Complex Numbers

Example Question #61 : Basic Operations With Complex Numbers

Example Question #65 : Basic Operations With Complex Numbers

Example Question #66 : Basic Operations With Complex Numbers

Example Question #67 : Basic Operations With Complex Numbers

Example Question #121 : Imaginary Numbers

All Algebra II Resources

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