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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #112 : Imaginary Numbers

Evaluate: $-2(i^{10}-i^{25})$

Possible Answers:

2+2i

2-2i

2-i

-2-2i

Correct answer:

2+2i

Explanation:

The value of $i=\sqrt{-1}$ . We can rewrite the expression as the product of exponents to simplify.

$-2(i^{10}-i^{25})=-2((i^{2})^5-(i^{5})^{5})$

Identify the values of i^2 and i^5 .

$i^2=i\cdot i=\sqrt{-1} \cdot\sqrt{-1} = -1$

$i^5 = i^2 \cdot i^2 \cdot i = (-1)(-1)i = i$

Replace the values and simplify.

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

The answer is: 2+2i

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Example Question #2071 : Mathematical Relationships And Basic Graphs

Solve: (i^2+2i^8)^2

Possible Answers:

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

Correct answer:

Explanation:

Evaluate the inner terms of the imaginary values.

The value of $i=\sqrt{-1}$ .

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

i^8 = (i^2)^4 = (-1)^4 =1

The expression becomes:

(-1+2(1))^2 = (-1+2)^2 = (1)^2 =1

The answer is:

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Example Question #114 : Imaginary Numbers

Express the following number in the form a+bi , where and are real numbers:

$\frac{17}{4-3i}$

Possible Answers:

$17+\frac{51}{4}i$

$\frac{68}{25}+\frac{51}{25}i$

$\frac{68}{25}-\frac{51}{25}i$

$\frac{68}{7}-\frac{51}{7}i$

Correct answer:

$\frac{68}{25}+\frac{51}{25}i$

Explanation:

Since there is a complex number in the form a+bi in the denominator of the given expression, multiply the numerator and the denominator of the expression by its complex conjugate:

$\frac{17}{4-3i}=(\frac{17}{4-3i})(\frac{4+3i}{4+3i})$

$=\frac{17(4+3i)}{(4-3i)(4+3i)}$

$=\frac{68+51i}{16-9i^2}$

$=\frac{68+51i}{16+9}$

$=\frac{68+51i}{25}$

$=\frac{68}{25}+\frac{51}{25}i$

This number is now expressed in the form a+bi , where $a=\frac{68}{25}$ and $b=\frac{51}{25}$ . Hence, the correct answer is $\frac{68}{25}+\frac{51}{25}i$ .

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Example Question #2071 : Mathematical Relationships And Basic Graphs

Evaluate: $2i^{243}-i^3$

Possible Answers:

-3i

-2i

Correct answer:

Explanation:

Recall that $i=\sqrt{-1}$ .

$i^3 = i^2\cdot i =(-1)i = -i$

Rewrite the power of 243 as a product of a power of three.

$2i^{243}-i^3 =2(i^{3})^{81}-i^3 =2(-i)^{81}-(-i)$

Simplify the expression.

2(-i)-(-i) = -2i+i = -i

The answer is:

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Example Question #116 : Imaginary Numbers

Determine the value of: $3i^{1000}$

Possible Answers:

-3i

Correct answer:

Explanation:

In order to determine the exact value of $i^{1000}$ , we will need to rewrite this exponent as a product of exponents.

$i^{1000} =( i^4)^{250}$

Recall that $i=\sqrt{-1}$ .

i^2=-1

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

$( i^4)^{250}= (1)^{250} = 1$

Therefore,

$3i^{1000} = 3(1) =3$

The answer is:

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Example Question #2071 : Mathematical Relationships And Basic Graphs

Add: $4i^4-8i^{20}-12i^2$

Possible Answers:

$\textup{The terms are simplified as is.}$

Correct answer:

Explanation:

The expression contain imaginary terms. The values can be simplified.

Recall that:

$i=\sqrt{-1}$

This indicates that:

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

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

Rewrite the expression. For powers higher than the given terms, we can rewrite that power as a product of exponents.

$4i^4-8i^{20}-12i^2 = 4(1)-8(i^4)^5-12(-1)$

4-8(1)^5+12 = 4-8+12 = 8

The answer is:

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Example Question #121 : Imaginary Numbers

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

Possible Answers:

-2-i

-102-i

-100+i

-2i

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 #2071 : Mathematical Relationships And Basic Graphs

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

Possible Answers:

-2i

2-i

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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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #112 : Imaginary Numbers

Example Question #2071 : Mathematical Relationships And Basic Graphs

Example Question #114 : Imaginary Numbers

Example Question #2071 : Mathematical Relationships And Basic Graphs

Example Question #116 : Imaginary Numbers

Example Question #2071 : Mathematical Relationships And Basic Graphs

Example Question #121 : Imaginary Numbers

Example Question #2071 : Mathematical Relationships And Basic Graphs

Example Question #61 : Basic Operations With Complex Numbers

Example Question #2071 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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