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Common Core: High School - Number and Quantity : Complex Number Form: CCSS.Math.Content.HSN-CN.A.1

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Example Questions

Example Question #1 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Simplifying the following.

$\sqrt{-9}\times\sqrt{18}$

Possible Answers:

$9i^2\sqrt{2}$

$9i\sqrt{2}$

$9\sqrt{2}$

$-9\sqrt{2}$

$-9i\sqrt{2}$

Correct answer:

$9i\sqrt{2}$

Explanation:

This question tests one's ability to perform arithmetic operations on complex numbers. Questions like this introduces and builds on the concept of complex numbers. Recall that a complex number by definition contains a negative square. In mathematical terms this is expressed as follows.

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

Performing arithmetic operations on complex numbers relies on the understanding of the various algebraic operations and properties (distributive, associative, and commutative properties) as well as the imaginary, complex number .

For the purpose of Common Core Standards, "know there is a complex number such that i^2=-1 , and every complex number has a form a+bi with and are reals", falls within the Cluster A of "perform arithmetic operations with complex numbers" (CCSS.MATH.CONTENT.HSF.CN.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Perform multiplication between the two terms.

$\sqrt{-9}\times\sqrt{18}$

Recall that multiplication between two radicand terms (terms under the square root sign), can be combined as one using the communicative property with multiplication.

$\\ \sqrt{a}\times \sqrt{b}=\sqrt{a\times b} \\ \sqrt{-9}\times\sqrt{18}=\sqrt{-9\times 18}$

Step 2: Factor the two term in the expression.

$\\=\sqrt{-9\times 18} \\=\sqrt{-1\times 9\times 9\times 2}$

Step 3: Pull out common terms that exists in the radicand.

Remember that when a number appears under the square root sign, one of the numbers can be brought out front and the other one is canceled out.

$\sqrt{a\times a}= a\sqrt{1}=a$

$\\=\sqrt{-1\times 9\times 9\times 2} \\=9\sqrt{-1\times 2}$

Step 4: Use the identity that $\sqrt{-1}=i$ .

$\\=9\sqrt{-1\times 2} \\=9i\sqrt{2}$

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Example Question #1 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Express $\sqrt{-9}$ as a pure imaginary number.

Possible Answers:

Correct answer:

Explanation:

A pure imaginary number is expressed as , where is a positive real number and represents the imaginary unit.

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Example Question #2 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Express $\sqrt{-45}$ as a pure imaginary number.

Possible Answers:

$3\sqrt{5}$

$3i\sqrt{5}$

$\sqrt{45}$

45i

Correct answer:

$3i\sqrt{5}$

Explanation:

A pure imaginary number is expressed as , where is a positive real number and represents the imaginary unit.

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Example Question #3 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Simplify:

$\sqrt{-225}$

Possible Answers:

$\sqrt{225}$

15i

Correct answer:

15i

Explanation:

$\sqrt{-225}=\sqrt{225}\cdot \sqrt{-1}$

$=15\cdot \sqrt{-1}$

=15i

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Example Question #4 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Simplify:

$\sqrt{-98}$

Possible Answers:

$-7\sqrt{2}$

$7\sqrt{2}$

$7i\sqrt{2}$

Correct answer:

$7i\sqrt{2}$

Explanation:

$\sqrt{-98}=\sqrt{98}\cdot \sqrt{-1}$

$=\sqrt{49}\cdot \sqrt{2}\cdot \sqrt{-1}$

$=7\sqrt{2}\cdot \sqrt{-1}$

$=7\sqrt{2}\cdot i$

$=7i\sqrt{2}$

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Example Question #5 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Simplify:

$\sqrt{-242}$

Possible Answers:

$-11\sqrt{2}$

$11\sqrt{2}$

$11i\sqrt{2}$

Correct answer:

$11i\sqrt{2}$

Explanation:

$\sqrt{-242}=\sqrt{242}\cdot \sqrt{-1}$

$=\sqrt{121}\cdot \sqrt{2}\cdot \sqrt{-1}$

$=11\cdot \sqrt{2}\cdot \sqrt{-1}$

$=11i\sqrt{2}$

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Example Question #1 : The Complex Number System

Simplify

$i^{2}$

Possible Answers:

$\sqrt{-1}$

Correct answer:

Explanation:

The powers of are:

$i=\sqrt{-1}$

$i^{2}=-1$

$i^{3}=-\sqrt{-1}$

$i^{4}=1$

This pattern continues for every successive four power of . Thus:

$i^{5}=\sqrt{-1}$

$i^{6}=-1$

$i^{7}=-\sqrt{-1}$

$i^{8}=1$

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Example Question #6 : The Complex Number System

Simplify:

$i^{3}$

Possible Answers:

$-\sqrt{-1}$

$\sqrt{-1}$

Correct answer:

$-\sqrt{-1}$

Explanation:

The powers of are:

$i=\sqrt{-1}$

$i^{2}=-1$

$i^{3}=-\sqrt{-1}$

$i^{4}=1$

This pattern continues for every successive four power of . Thus:

$i^{5}=\sqrt{-1}$

$i^{6}=-1$

$i^{7}=-\sqrt{-1}$

$i^{8}=1$

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Example Question #9 : The Complex Number System

Simplify:

$i^{7}$

Possible Answers:

$-\sqrt{-1}$

Correct answer:

$-\sqrt{-1}$

Explanation:

The powers of are:

$i=\sqrt{-1}$

$i^{2}=-1$

$i^{3}=-\sqrt{-1}$

$i^{4}=1$

This pattern continues for every successive four power of . Thus:

$i^{5}=\sqrt{-1}$

$i^{6}=-1$

$i^{7}=-\sqrt{-1}$

$i^{8}=1$

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Example Question #10 : The Complex Number System

Simplify:

$i^{50}$

Possible Answers:

50i

$\sqrt{-1}$

Correct answer:

Explanation:

The powers of are:

$i=\sqrt{-1}$

$i^{2}=-1$

$i^{3}=-\sqrt{-1}$

$i^{4}=1$

This pattern continues for every successive four power of . Thus:

$i^{5}=\sqrt{-1}$

$i^{6}=-1$

$i^{7}=-\sqrt{-1}$

$i^{8}=1$

To simplify to a larger power, simply break it into $i^{4}$ terms, as these simplify to 1.

$i^{50}=\left( i^{4}\right)^{12}\cdot i^{2}$

$=1\cdot i^{2}$

$=i^{2}$

=-1

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Common Core: High School - Number and Quantity : Complex Number Form: CCSS.Math.Content.HSN-CN.A.1

Study concepts, example questions & explanations for Common Core: High School - Number and Quantity

All Common Core: High School - Number and Quantity Resources

Example Questions

Example Question #1 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Example Question #1 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Example Question #2 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Example Question #3 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Example Question #4 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Example Question #5 : Complex Number Form: Ccss.Math.Content.Hsn Cn.A.1

Example Question #1 : The Complex Number System

Example Question #6 : The Complex Number System

Example Question #9 : The Complex Number System

Example Question #10 : The Complex Number System

All Common Core: High School - Number and Quantity Resources

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