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Common Core: High School - Number and Quantity : High School: Number and Quantity

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

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All Common Core: High School - Number and Quantity Resources

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Example Questions

Example Question #4 : The Complex Number System

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 : The Complex Number System

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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Example Question #11 : The Complex Number System

Simplify:

$i^{243}$

Possible Answers:

$-\sqrt{-1}$

$\sqrt{-1}$

243i

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:

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

$i^{6}=-1$

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

$i^{8}=1$

For very large powers, we can begin by dividing the exponent by 4:

$243\div 4=60\textup{ r}3$

That means that we can break the exponent down as follows:

$i^{243}=\left ( i^{4} \right )^{60}\cdot i^{3}$

$=1\cdot i^{3}$

$=i^{3}$

$=-\sqrt{-1}$

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Example Question #41 : High School: Number And Quantity

Simplify:

$\sqrt{-\frac{169}{225}}$

Possible Answers:

$-\frac{13}{15}$

$\frac{13}{15}i$

$-\frac{169}{225}i$

Correct answer:

$\frac{13}{15}i$

Explanation:

$\sqrt{-\frac{169}{225}}=\sqrt{\frac{169}{225}}\cdot \sqrt{-1}$

$=\frac{\sqrt{169}}{\sqrt{225}}\cdot\sqrt{-1}$

$=\frac{13}{15}\cdot \sqrt{-1}$

$=\frac{13}{15} i$

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Example Question #1 : Complex I Relationship And The Commutative, Associative, And Distributive Properties: Ccss.Math.Content.Hsn Cn.A.2

What does the following equal?

(7+2i)-(6-4i)

Possible Answers:

-1-6i

1+6i

1-6i

6+i

-1+6i

Correct answer:

1+6i

Explanation:

This question tests ones ability to recognize that when adding and subtracting complex numbers, the imaginary portions act like a variable during algebraic operations. When adding or subtracting complex numbers recall that it is the combination of like terms. The liked terms of complex numbers include the real portion of the complex number (the 's), and the imaginary portion (the 's) given the expression,

$(a+bi)\pm(a_1+b_1i)$ .

For the purpose of Common Core Standards, "use the relation i^2=-1 , and the commutative, associate, and distributive properties to add, subtract, and multiply complex numbers", 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: Distribute the negative to each term within the second set of parentheses.

(7+2i)-(6-4i)

7+2i-6-(-4i)

Recall that subtracting a negative number means adding a positive.

7+2i-6+4i

Step 2: Identify like terms.

Real portion: 7-6=1

Imaginary portion: 2i+4i=6i

Note: treat as a variable.

Step 3: Combine like terms into one expression.

1+6i

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Example Question #42 : High School: Number And Quantity

Simplify.

$2i\cdot 3i$

Possible Answers:

Correct answer:

Explanation:

$2i\cdot 3i=\left ( 2\cdot 3 \right )\left ( i\cdot i \right )$

$=6i^{2}$

$=6\cdot -1$

=-6

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Common Core: High School - Number and Quantity : High School: Number and Quantity

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 #4 : The Complex Number System

Example Question #5 : The Complex Number System

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

Example Question #11 : The Complex Number System

Example Question #41 : High School: Number And Quantity

Example Question #1 : Complex I Relationship And The Commutative, Associative, And Distributive Properties: Ccss.Math.Content.Hsn Cn.A.2

Example Question #42 : High School: Number And Quantity

All Common Core: High School - Number and Quantity Resources

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