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Algebra II : Number Sets

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Example Question #11 : Number Sets

True or false:

The set $\left \{ i^{1}, i^{2}, i^{3}, i^{4}, i^{5}\right \}$ comprises only imaginary numbers.

Possible Answers:

True

False

Correct answer:

False

Explanation:

Any even power of the imaginary unit is a real number. For example,

$i^{2}= -1$ from the definition of as the principal square root of .

Also, from the Power of a Power Property,

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

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Example Question #12 : Number Sets

What is $A\cap B$ ?

$A=\{1,3,5,8,9,11,15\}$

$B=\{1,2,4,6,9,11\}$

Possible Answers:

$\{9,11\}$

$\{1,11\}$

$\{1,9\}$

$\{1,9,11\}$

$\{1,2,3,4,5,6,8,9,11,15\}$

Correct answer:

$\{1,9,11\}$

Explanation:

$A\cap B$ or A intersect B means what A and B have in common.

$A=\{{\color{Red} 1},3,5,8,{\color{Red} 9,11},15\}$

$B=\{{\color{Red} 1},2,4,6,{\color{Red} 9,11}\}$

In this case both A and B have the numbers 1, 9, and 11.

$A\cap B=\{1,9,11\}$

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Example Question #12 : Number Sets

True or false:

The following set comprises only imaginary numbers:

$\left \{ i^{7}, i^{9}, i^{11}, i^{13}, i^{15}\right \}$

Possible Answers:

False

True

Correct answer:

True

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

Every element in the set $\left \{ i^{7}, i^{9}, i^{11}, i^{13}, i^{15}\right \}$ is equal to raised to an odd-numbered power, so when each exponent is divided by 4, the remainder will be either 1 or 3. Therefore, each element is equal to either or . Consequently, the set includes only imaginary numbers.

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Example Question #12 : How To Find A Solution Set

Set A is composed of all multiples of 4 that are that are less than the square of 7. Set B includes all multiples of 6 that are greater than 0. How many numbers are found in both set A and set B?

Possible Answers:

Correct answer:

Explanation:

Start by making a list of the multiples of 4 that are smaller than the square of 7. When 7 is squared, it equals 49; thus, we can compose the following list:

$\text{4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, and 48}$

Next, make a list of all the multiples of 6 that are greater than 0. Since we are looking for shared multiples, stop after 48 because numbers greater than 48 will not be included in set A. The biggest multiple of 4 smaller that is less than 49 is 48; therefore, do not calculate multiples of 6 greater than 48.

$\text{6, 12, 18, 24, 30, 36, 42, and 48}$

Finally, count the number of multiples found in both sets. Both sets include the following numbers:

$\text{12, 24, 36, and 48}$

The correct answer is 4 numbers.

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Algebra II : Number Sets

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Number Sets

Example Question #12 : Number Sets

Example Question #12 : Number Sets

Example Question #12 : How To Find A Solution Set

All Algebra II Resources

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