Question

simplify the expression. write your answer as a complex number.

(2sqrt{25} + sqrt{-16})

• (10 + 4i)
• (10 - 4i)
• (10 + 8i)
• (10 - 8i)

question 4 (multiple choice worth 6 points)
(01.07r mc)

multiply ((6 - 7i)(3 - 6i)).

• (18 - 57i + 42i^2)
• (18 + 42i^2)
• (60 - 57i)
• (-24 - 57i)

Explanation:

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<reasoning>

Simplify the first expression

We simplify the real and imaginary parts of the expression \(2\sqrt{25} + \sqrt{-16}\).
First, evaluate the real square root:
\[2\sqrt{25} = 2 \cdot 5 = 10\]
Next, evaluate the imaginary square root using \(i = \sqrt{-1}\):
\[\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\]
Combining these terms gives:
\[10 + 4i\]

Multiply the second expression

We expand the product of two complex numbers \((6 - 7i)(3 - 6i)\) using the FOIL method.
Multiply the terms:
\[(6)(3) + (6)(-6i) + (-7i)(3) + (-7i)(-6i)\]
\[= 18 - 36i - 21i + 42i^2\]

Simplify the product

Combine the imaginary terms and substitute \(i^2 = -1\):
\[18 - 57i + 42(-1)\]
\[= 18 - 57i - 42\]
Combine the real constants:
\[(18 - 42) - 57i = -24 - 57i\]
</reasoning>

<answer>

Question 1

<mcq-correct>(A) \(10 + 4i\)</mcq-correct>
<mcq-option>(B) \(10 - 4i\)</mcq-option>
<mcq-option>(C) \(10 + 8i\)</mcq-option>
<mcq-option>(D) \(10 - 8i\)</mcq-option>

Question 2

<mcq-option>(A) \(18 - 57i + 42i^2\)</mcq-option>
<mcq-option>(B) \(18 + 42i^2\)</mcq-option>
<mcq-option>(C) \(60 - 57i\)</mcq-option>
<mcq-correct>(D) \(-24 - 57i\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Simplify the first expression

We simplify the real and imaginary parts of the expression \(2\sqrt{25} + \sqrt{-16}\).
First, evaluate the real square root:
\[2\sqrt{25} = 2 \cdot 5 = 10\]
Next, evaluate the imaginary square root using \(i = \sqrt{-1}\):
\[\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\]
Combining these terms gives:
\[10 + 4i\]

Multiply the second expression

We expand the product of two complex numbers \((6 - 7i)(3 - 6i)\) using the FOIL method.
Multiply the terms:
\[(6)(3) + (6)(-6i) + (-7i)(3) + (-7i)(-6i)\]
\[= 18 - 36i - 21i + 42i^2\]

Simplify the product

Combine the imaginary terms and substitute \(i^2 = -1\):
\[18 - 57i + 42(-1)\]
\[= 18 - 57i - 42\]
Combine the real constants:
\[(18 - 42) - 57i = -24 - 57i\]
</reasoning>

<answer>

Question 1

<mcq-correct>(A) \(10 + 4i\)</mcq-correct>
<mcq-option>(B) \(10 - 4i\)</mcq-option>
<mcq-option>(C) \(10 + 8i\)</mcq-option>
<mcq-option>(D) \(10 - 8i\)</mcq-option>

Question 2

<mcq-option>(A) \(18 - 57i + 42i^2\)</mcq-option>
<mcq-option>(B) \(18 + 42i^2\)</mcq-option>
<mcq-option>(C) \(60 - 57i\)</mcq-option>
<mcq-correct>(D) \(-24 - 57i\)</mcq-correct>
</answer>

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