Question

subject: math
class: ss1 time
instruction: answer all que

1. which of the following is a surd?

a. \\(\sqrt{4}\\)
b. \\(\sqrt{9}\\)
c. \\(\sqrt{7}\\)
d. \\(\sqrt{16}\\)

1. simplify \\(\sqrt{45}\\).

a. \\(3\sqrt{5}\\)
b. \\(5\sqrt{3}\\)
c. \\(9\sqrt{2}\\)
d. \\(2\sqrt{5}\\)

1. what is the conjugate of \\(\sqrt{2} - \sqrt{5}\\)?

a. \\(\sqrt{2} + \sqrt{5}\\)
b. \\(2\sqrt{2} - \sqrt{5}\\)
c. \\(\sqrt{5} - \sqrt{2}\\)
d. \\(\sqrt{5} + \sqrt{2}\\)

1. simplify \\(\sqrt{32} + 3\sqrt{8}\\).

a. \\(10\sqrt{2}\\)
b. \\(5\sqrt{2}\\)
c. \\(8\sqrt{3}\\)
d. \\(12\sqrt{3}\\)

1. evaluate \\((2\sqrt{3} - \sqrt{5})(2\sqrt{3} + \sqrt{5})\\).

a. 7

Explanation:

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<reasoning>

Identify the surd in Question 1

\[

$$\begin{aligned} &\sqrt{4} = 2 \quad (\text{Rational})\\ &\sqrt{9} = 3 \quad (\text{Rational})\\ &\sqrt{7} \approx 2.6457... \quad (\text{Irrational / Surd})\\ &\sqrt{16} = 4 \quad (\text{Rational}) \end{aligned}$$

\]

Simplify the radical in Question 2

\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
\]

Find the conjugate in Question 3

\[
\text{Conjugate of } (\sqrt{2} - \sqrt{5}) = \sqrt{2} + \sqrt{5}
\]

Simplify the sum in Question 4

\[

$$\begin{aligned} &\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\\ &3\sqrt{8} = 3\sqrt{4 \times 2} = 3 \times 2\sqrt{2} = 6\sqrt{2}\\ &\sqrt{32} + 3\sqrt{8} = 4\sqrt{2} + 6\sqrt{2} = 10\sqrt{2} \end{aligned}$$

\]

Evaluate the product in Question 5

\[

$$\begin{aligned} (2\sqrt{3} - \sqrt{5})(2\sqrt{3} + \sqrt{5}) &= (2\sqrt{3})^2 - (\sqrt{5})^2\\ &= 4 \times 3 - 5\\ &= 12 - 5\\ &= 7 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>A. \(\sqrt{4}\)</mcq-option>
<mcq-option>B. \(\sqrt{9}\)</mcq-option>
<mcq-correct>C. \(\sqrt{7}\)</mcq-correct>
<mcq-option>D. \(\sqrt{16}\)</mcq-option>

Question 2

<mcq-correct>A. \(3\sqrt{5}\)</mcq-correct>
<mcq-option>B. \(5\sqrt{3}\)</mcq-option>
<mcq-option>C. \(9\sqrt{2}\)</mcq-option>
<mcq-option>D. \(2\sqrt{5}\)</mcq-option>

Question 3

<mcq-correct>A. \(\sqrt{2} + \sqrt{5}\)</mcq-correct>
<mcq-option>B. \(2\sqrt{2} - \sqrt{5}\)</mcq-option>
<mcq-option>C. \(\sqrt{5} - \sqrt{2}\)</mcq-option>
<mcq-option>D. \(\sqrt{5} + \sqrt{2}\)</mcq-option>

Question 4

<mcq-correct>A. \(10\sqrt{2}\)</mcq-correct>
<mcq-option>B. \(5\sqrt{2}\)</mcq-option>
<mcq-option>C. \(8\sqrt{3}\)</mcq-option>
<mcq-option>D. \(12\sqrt{3}\)</mcq-option>

Question 5

<mcq-correct>A. 7</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the surd in Question 1

\[

$$\begin{aligned} &\sqrt{4} = 2 \quad (\text{Rational})\\ &\sqrt{9} = 3 \quad (\text{Rational})\\ &\sqrt{7} \approx 2.6457... \quad (\text{Irrational / Surd})\\ &\sqrt{16} = 4 \quad (\text{Rational}) \end{aligned}$$

\]

Simplify the radical in Question 2

\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
\]

Find the conjugate in Question 3

\[
\text{Conjugate of } (\sqrt{2} - \sqrt{5}) = \sqrt{2} + \sqrt{5}
\]

Simplify the sum in Question 4

\[

$$\begin{aligned} &\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\\ &3\sqrt{8} = 3\sqrt{4 \times 2} = 3 \times 2\sqrt{2} = 6\sqrt{2}\\ &\sqrt{32} + 3\sqrt{8} = 4\sqrt{2} + 6\sqrt{2} = 10\sqrt{2} \end{aligned}$$

\]

Evaluate the product in Question 5

\[

$$\begin{aligned} (2\sqrt{3} - \sqrt{5})(2\sqrt{3} + \sqrt{5}) &= (2\sqrt{3})^2 - (\sqrt{5})^2\\ &= 4 \times 3 - 5\\ &= 12 - 5\\ &= 7 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>A. \(\sqrt{4}\)</mcq-option>
<mcq-option>B. \(\sqrt{9}\)</mcq-option>
<mcq-correct>C. \(\sqrt{7}\)</mcq-correct>
<mcq-option>D. \(\sqrt{16}\)</mcq-option>

Question 2

<mcq-correct>A. \(3\sqrt{5}\)</mcq-correct>
<mcq-option>B. \(5\sqrt{3}\)</mcq-option>
<mcq-option>C. \(9\sqrt{2}\)</mcq-option>
<mcq-option>D. \(2\sqrt{5}\)</mcq-option>

Question 3

<mcq-correct>A. \(\sqrt{2} + \sqrt{5}\)</mcq-correct>
<mcq-option>B. \(2\sqrt{2} - \sqrt{5}\)</mcq-option>
<mcq-option>C. \(\sqrt{5} - \sqrt{2}\)</mcq-option>
<mcq-option>D. \(\sqrt{5} + \sqrt{2}\)</mcq-option>

Question 4

<mcq-correct>A. \(10\sqrt{2}\)</mcq-correct>
<mcq-option>B. \(5\sqrt{2}\)</mcq-option>
<mcq-option>C. \(8\sqrt{3}\)</mcq-option>
<mcq-option>D. \(12\sqrt{3}\)</mcq-option>

Question 5

<mcq-correct>A. 7</mcq-correct>
</answer>

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