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Algebra II : Number Theory

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Example Questions

Example Question #61 : Number Theory

Write the following expression in the standard form for a complex number

i(3+7i)(-4+2i)

Possible Answers:

22-26i

22-2i

22+2i

$-12i-22i^{2}+14i^{^{3}}$

-26i+22

Correct answer:

22-26i

Explanation:

i(3+7i)(-4+2i)

Multiply Binomials ( you may use the FOIL method)

$i(-12+6i-28i+14i^{^{2}})$

We know that $i^{2}=-1$ , so we replace $i^{2}$ with

i(-12+6i-28i-14)

combine like terms

i(-26-22i)

Distribute the i

$-26i - 22i^{2}$

We know that $i^{2}=-1$ , so we replace $i^{2}$ with

-26i+22

Swich to standard form

22-26i

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Example Question #21 : Irrational Numbers

What is the sum of (7-2i) and -(-4+8i) ?

Possible Answers:

11+6i

3+6i

11-10i

3-2i

28-16i

Correct answer:

11-10i

Explanation:

(7-2i)+-(-4+8i)

Distribute the negative

7-2i+4-8i

Combine like terms

11-10i

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Example Question #22 : Irrational Numbers

Add and combine: $\frac{\pi}{e}+\frac{\sqrt2}{\pi}$

Possible Answers:

$\frac{e\pi^2+e\sqrt2}{e\pi}$

$\frac{\sqrt{\pi^2+2e}}{e\pi}$

$\frac{\pi^2+\sqrt2}{\pi}$

$\frac{\pi^2+\sqrt{2e}}{e\pi}$

$\frac{\pi^2+e\sqrt2}{e\pi}$

Correct answer:

$\frac{\pi^2+e\sqrt2}{e\pi}$

Explanation:

To simplify the irrational numbers as a single fraction, we will need a common denominator by multiplying the denominators together.

The term $e\pi$ is the common denominator. Convert the fractions.

$\frac{\pi}{e}+\frac{\sqrt2}{\pi}=\frac{\pi^2}{e\pi}+\frac{e\sqrt2}{e\pi} = \frac{\pi^2+e\sqrt2}{e\pi}$

The answer is: $\frac{\pi^2+e\sqrt2}{e\pi}$

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Example Question #23 : Irrational Numbers

Which of the following is considered an irrational number?

Possible Answers:

$\sqrt{144}$

$0.\bar{6}$

$\sqrt{30}$

ln(e)

3.14159

Correct answer:

$\sqrt{30}$

Explanation:

The irrational numbers do not have a representation of a ratio between two numbers. They cannot be expressed by a fraction.

Repeating decimal numbers are not irrational because they can be rewritten as a fraction.

For instance: $0.\bar{3} = \frac{1}{3}$

The number 3.14159 may represent the short version of $\pi$ , but is not irrational, because 3.14159 is a fixed number and be rewritten as a ratio between two numbers.

The answer is: $\sqrt{30}$

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Example Question #61 : Number Theory

$What\ is\ the\ complex\ conjugate\ for\ 3-\sqrt{2}\ ?$

Possible Answers:

$3-\sqrt{2}$

$9-\sqrt{4}$

$3+\sqrt{2}$

$9+\sqrt{4}$

Correct answer:

$3+\sqrt{2}$

Explanation:

The complex conjugate for an irrational binomial number with a radical is simply the original with the sign of the radical changed.

$When\ 3-\sqrt{2}\ is\ multipled\ by\ 3+\sqrt{2}\ this\ will\ produce\ a\ rational\ number.$

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Example Question #62 : Number Theory

Which of the below is a rational number?

Possible Answers:

$5-2\pi$

$\sqrt{225}$

$\sqrt{117}$

$\sqrt{8}$

Correct answer:

$\sqrt{225}$

Explanation:

$\sqrt{225}\ when\ simplified\ becomes\ 15\ which\ is\ a\ rational\ number.$

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Example Question #5122 : Algebra Ii

$5-\sqrt{2}\ multipled\ by\ which\ of\ the\ below\ will\ produce\ a\ rational\ number?$

Possible Answers:

$\sqrt{5-\sqrt{2}}$

$5+\sqrt{2}$

$5-\sqrt{2}$

$\sqrt{2}$

Correct answer:

$5+\sqrt{2}$

Explanation:

$An\ irrational\ binomial\ must\ be\ multiplied\ by\ its'\ complex\ conjugate\ in\ order\ to\ produce\ a\ rational\ number.$

$The\ complex\ conjugate\ for\ 5-\sqrt{2}\ is\ 5+\sqrt{2}$

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Example Question #68 : Number Theory

Which of the following is an irrational number?

Possible Answers:

sin(270)

3.14159

$\sqrt8$

$\sqrt{169}$

$i^{50}$

Correct answer:

$\sqrt8$

Explanation:

Irrational numbers cannot be expressed as a ratio of two numbers. They can be decimal numbers that go on forever without repeating.

Do not mix fixed numbers with symbolic values such as 3.14159 and $\pi$ .

The value of the sine angle sin(270) is , which is not irrational.

Square root numbers and complex numbers might not necessary be irrational after their simplified form.

$i^{50}= (i^2)^{25} = (-1)^{25} = -1$

$\sqrt8 = 2\sqrt2$

The correct answer is: $\sqrt8$

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Example Question #69 : Number Theory

Which of the following numbers are irrational?

Possible Answers:

$\sqrt6$

3.14159

$\50\%$

$\sqrt[3]{8}$

$\frac{199}{100}$

Correct answer:

$\sqrt6$

Explanation:

Irrational numbers are numbers that cannot be rewritten as a fraction of two numbers.

Be careful with numbers that may look as though they are irrational, such as 3.14159 , but is rational since this number is finite and can be expressed as a fraction.

Irrational numbers cannot include continuous numbers such as $\pi$ , and some radical numbers such as $\sqrt2$ .

Some of the radicals in the answers are not fully simplified.

The term $\sqrt[3]{8}$ can be simplified to a whole number two, which means that this is a rational number.

The correct answer is: $\sqrt6$

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Example Question #63 : Number Theory

Which of the below is NOT an irrational number?

Possible Answers:

$\sqrt{-5}$

$10-\pi$

$\sqrt{16}-\sqrt{49}$

$\sqrt{12}-3$

Correct answer:

$\sqrt{16}-\sqrt{49}$

Explanation:

The correct choice involves two perfect squares which makes each number a rational number:

$\sqrt{16}-\sqrt{49}=4-7=-3\ which\ is\ a\ rational\ number$

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Algebra II : Number Theory

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #61 : Number Theory

Example Question #21 : Irrational Numbers

Example Question #22 : Irrational Numbers

Example Question #23 : Irrational Numbers

Example Question #61 : Number Theory

Example Question #62 : Number Theory

Example Question #5122 : Algebra Ii

Example Question #68 : Number Theory

Example Question #69 : Number Theory

Example Question #63 : Number Theory

Which of the below is NOT an irrational number?

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