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

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Question

Simplifying the following.

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

Answer

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 , and every complex number has a form 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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Question

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

Answer

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

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Question

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

Answer

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

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Question

Simplify:

$\sqrt{-225}$

Answer

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

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

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Question

Simplify:

$\sqrt{-98}$

Answer

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

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

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

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

$=7i\sqrt{2}$

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Question

Simplify:

$\sqrt{-242}$

Answer

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

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

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

$=11i\sqrt{2}$

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Question

Simplify

$i^{2}$

Answer

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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Question

Simplify:

$i^{3}$

Answer

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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Question

Simplify:

$i^{7}$

Answer

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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Question

Simplify:

$i^{50}$

Answer

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.

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

$=1\cdot i^{2}$

$=i^{2}$

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Question

Simplify:

$i^{243}$

Answer

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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Question

Simplify:

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

Answer

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

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

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

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

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Question

Simplifying the following.

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

Answer

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 , and every complex number has a form 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 $\sqrt{-1}=i$ .

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

Simplify:

$\sqrt{-225}$

Answer

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

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

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Question

Simplify:

$\sqrt{-98}$

Answer

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

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

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

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

$=7i\sqrt{2}$

Compare your answer with the correct one above

Question

Simplify:

$\sqrt{-242}$

Answer

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

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

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

$=11i\sqrt{2}$

Compare your answer with the correct one above

Question

Simplify

$i^{2}$

Answer

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:

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

$i^{6}=-1$

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

$i^{8}=1$

Compare your answer with the correct one above

Question

Simplify:

$i^{3}$

Answer

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:

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

$i^{6}=-1$

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

$i^{8}=1$

Compare your answer with the correct one above

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