Question

simplify \\(-\sqrt{-48}\\).

a) \\(-4\sqrt{3}i\\)
b) \\(4\sqrt{3}i\\)
c) \\(-4\sqrt{3}i\\)
d) \\(4\sqrt{3}\\)

question 14 (5 points)
find all solution(s) of the quadratic equation \\(x^2 - 100 = 0\\).

a) \\(x = 10\\)
b) \\(x = 10, -10\\)
c) \\(x = 100, -100\\)
d) \\(x = -10\\)

question 15 (5 points)
write \\(\sqrt{12} + \sqrt{-9}\\) as a complex number.

a) \\(5\sqrt{3}i\\)
b) \\(2\sqrt{3} + 3i\\)
c) \\(2\sqrt{3} - 3i\\)
d) \\(2\sqrt{3} + 9i\\)

Explanation:

Simplify the radical expression

Using the Simplifying Radicals with Negative Radicands knowledge point, we simplify \(-\sqrt{-48}\).
First, express the negative radicand using the imaginary unit \(i = \sqrt{-1}\):

$$ -\sqrt{-48} = -\sqrt{48 \cdot (-1)} = -\sqrt{48} \cdot i $$

Next, factor out the largest perfect square from \(48\), which is \(16\):

$$ -\sqrt{16 \cdot 3} \cdot i = -4\sqrt{3}i $$

Solve the quadratic equation

Using the Square Root Principle knowledge point

$$
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Write the sum as a complex number

Using the Complex Number Representation knowledge point, we simplify \(\sqrt{12} + \sqrt{-9}\).
First, simplify the real radical term \(\sqrt{12}\):

$$ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} $$

Next, simplify the imaginary radical term \(\sqrt{-9}\):

$$ \sqrt{-9} = \sqrt{9 \cdot (-1)} = 3i $$

Combine the real and imaginary parts to write the final complex number:

$$ 2\sqrt{3} + 3i $$

Answer:

Question 13

• (A) \(-4\sqrt{5}\)
• (B) \(4\sqrt{3}i\)
• (C) \(-4\sqrt{3}i\) (Correct answer)
• (D) \(4\sqrt{5}\)

Question 14

• (A) \(x = 10\)
• (B) \(x = 10, -10\) (Correct answer)
• (C) \(x = 100, -100\)
• (D) \(x = -10\)

Question 15

• (A) \(5\sqrt{3}i\)
• (B) \(2\sqrt{3} + 3i\) (Correct answer)
• (C) \(2\sqrt{3} - 3i\)
• (D) \(2\sqrt{3} + 9i\)