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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

6 Diagnostic Tests 49 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #4 : The Complex Number System

Simplify:

$\sqrt{-98}$ $\displaystyle \sqrt{-98}$

Possible Answers:

$7\sqrt{2}$ $\displaystyle 7\sqrt{2}$

$-7\sqrt{2}$ $\displaystyle -7\sqrt{2}$

$7i\sqrt{2}$ $\displaystyle 7i\sqrt{2}$

$\displaystyle 7i$

Correct answer:

$7i\sqrt{2}$ $\displaystyle 7i\sqrt{2}$

Explanation:

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

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

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

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

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

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Example Question #5 : The Complex Number System

Simplify:

$\sqrt{-242}$ $\displaystyle \sqrt{-242}$

Possible Answers:

$-11\sqrt{2}$ $\displaystyle -11\sqrt{2}$

$11\sqrt{2}$ $\displaystyle 11\sqrt{2}$

$11i\sqrt{2}$ $\displaystyle 11i\sqrt{2}$

$\displaystyle -13$

Correct answer:

$11i\sqrt{2}$ $\displaystyle 11i\sqrt{2}$

Explanation:

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

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

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

$=11i\sqrt{2}$ $\displaystyle =11i\sqrt{2}$

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Example Question #3 : The Complex Number System

Simplify

$i^{2}$ $\displaystyle i^{2}$

Possible Answers:

$\displaystyle 2$

$\displaystyle 1$

$\displaystyle -1$

$\sqrt{-1}$ $\displaystyle \sqrt{-1}$

Correct answer:

$\displaystyle -1$

Explanation:

The powers of $\displaystyle i$ are:

$i=\sqrt{-1}$ $\displaystyle i=\sqrt{-1}$

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

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

$i^{4}=1$ $\displaystyle i^{4}=1$

This pattern continues for every successive four power of $\displaystyle i$. Thus:

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

$i^{6}=-1$ $\displaystyle i^{6}=-1$

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

$i^{8}=1$ $\displaystyle i^{8}=1$

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Example Question #4 : The Complex Number System

Simplify:

$i^{3}$ $\displaystyle i^{3}$

Possible Answers:

$\displaystyle -1$

$\displaystyle 2$

$\sqrt{-1}$ $\displaystyle \sqrt{-1}$

$-\sqrt{-1}$ $\displaystyle -\sqrt{-1}$

Correct answer:

$-\sqrt{-1}$ $\displaystyle -\sqrt{-1}$

Explanation:

The powers of $\displaystyle i$ are:

$i=\sqrt{-1}$ $\displaystyle i=\sqrt{-1}$

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

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

$i^{4}=1$ $\displaystyle i^{4}=1$

This pattern continues for every successive four power of $\displaystyle i$. Thus:

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

$i^{6}=-1$ $\displaystyle i^{6}=-1$

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

$i^{8}=1$ $\displaystyle i^{8}=1$

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Example Question #5 : The Complex Number System

Simplify:

$i^{7}$ $\displaystyle i^{7}$

Possible Answers:

$-\sqrt{-1}$ $\displaystyle -\sqrt{-1}$

$\displaystyle 1$

$\displaystyle 2$

$\displaystyle -1$

Correct answer:

$-\sqrt{-1}$ $\displaystyle -\sqrt{-1}$

Explanation:

The powers of $\displaystyle i$ are:

$i=\sqrt{-1}$ $\displaystyle i=\sqrt{-1}$

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

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

$i^{4}=1$ $\displaystyle i^{4}=1$

This pattern continues for every successive four power of $\displaystyle i$. Thus:

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

$i^{6}=-1$ $\displaystyle i^{6}=-1$

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

$i^{8}=1$ $\displaystyle i^{8}=1$

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Example Question #6 : The Complex Number System

Simplify:

$i^{50}$ $\displaystyle i^{50}$

Possible Answers:

50i $\displaystyle 50i$

$\displaystyle 1$

$\sqrt{-1}$ $\displaystyle \sqrt{-1}$

$\displaystyle -1$

Correct answer:

$\displaystyle -1$

Explanation:

The powers of $\displaystyle i$ are:

$i=\sqrt{-1}$ $\displaystyle i=\sqrt{-1}$

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

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

$i^{4}=1$ $\displaystyle i^{4}=1$

This pattern continues for every successive four power of $\displaystyle i$. Thus:

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

$i^{6}=-1$ $\displaystyle i^{6}=-1$

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

$i^{8}=1$ $\displaystyle i^{8}=1$

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

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

$=1\cdot i^{2}$ $\displaystyle =1\cdot i^{2}$

$=i^{2}$ $\displaystyle =i^{2}$

=-1 $\displaystyle =-1$

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Example Question #11 : The Complex Number System

Simplify:

$i^{243}$ $\displaystyle i^{243}$

Possible Answers:

243i $\displaystyle 243i$

$\displaystyle -1$

$\sqrt{-1}$ $\displaystyle \sqrt{-1}$

$-\sqrt{-1}$ $\displaystyle -\sqrt{-1}$

Correct answer:

$-\sqrt{-1}$ $\displaystyle -\sqrt{-1}$

Explanation:

The powers of $\displaystyle i$ are:

$i=\sqrt{-1}$ $\displaystyle i=\sqrt{-1}$

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

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

$i^{4}=1$ $\displaystyle i^{4}=1$

This pattern continues for every successive four power of $\displaystyle i$. Thus:

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

$i^{6}=-1$ $\displaystyle i^{6}=-1$

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

$i^{8}=1$ $\displaystyle i^{8}=1$

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

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

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

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

$=1\cdot i^{3}$ $\displaystyle =1\cdot i^{3}$

$=i^{3}$ $\displaystyle =i^{3}$

$=-\sqrt{-1}$ $\displaystyle =-\sqrt{-1}$

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Example Question #11 : The Complex Number System

Simplify:

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

Possible Answers:

$\displaystyle 3i$

$-\frac{13}{15}$ $\displaystyle -\frac{13}{15}$

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

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

Correct answer:

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

Explanation:

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

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

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

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

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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?

$\displaystyle (7+2i)-(6-4i)$

Possible Answers:

-1-6i $\displaystyle -1-6i$

1+6i $\displaystyle 1+6i$

1-6i $\displaystyle 1-6i$

6+i $\displaystyle 6+i$

-1+6i $\displaystyle -1+6i$

Correct answer:

1+6i $\displaystyle 1+6i$

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 $\displaystyle a$'s), and the imaginary portion (the $\displaystyle b$'s) given the expression,

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

For the purpose of Common Core Standards, "use the relation i^2=-1 $\displaystyle i^2=-1$, 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.

$\displaystyle (7+2i)-(6-4i)$

$\displaystyle 7+2i-6-(-4i)$

Recall that subtracting a negative number means adding a positive.

$\displaystyle 7+2i-6+4i$

Step 2: Identify like terms.

Real portion: 7-6=1 $\displaystyle 7-6=1$

Imaginary portion: $\displaystyle 2i+4i=6i$

Note: treat $\displaystyle i$ as a variable.

Step 3: Combine like terms into one expression.

1+6i $\displaystyle 1+6i$

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Example Question #12 : The Complex Number System

Simplify.

$2i\cdot 3i$ $\displaystyle 2i\cdot 3i$

Possible Answers:

$\displaystyle 6i$

$\displaystyle 5i$

$\displaystyle 6$

$\displaystyle -6$

Correct answer:

$\displaystyle -6$

Explanation:

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

$=6i^{2}$ $\displaystyle =6i^{2}$

$=6\cdot -1$ $\displaystyle =6\cdot -1$

=-6 $\displaystyle =-6$

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All Common Core: High School - Number and Quantity Resources

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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 #3 : The Complex Number System

Example Question #4 : The Complex Number System

Example Question #5 : The Complex Number System

Example Question #6 : The Complex Number System

Example Question #11 : The Complex Number System

Example Question #11 : The Complex Number System

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

Example Question #12 : The Complex Number System

All Common Core: High School - Number and Quantity Resources

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