Surds and Indices

(6) If \( \sqrt{x} = 5 \) and \( \sqrt{y} = 3 \), what is the value of \( \sqrt{xy} \)?

• (a) 15
• (b) 2
• (c) 8
• (d) 4

Answer

Correct Answer: (a) 15
Given that \( \sqrt{x} = 5 \) and \( \sqrt{y} = 3 \), we can find \( \sqrt{xy} \) as follows:
\[ \sqrt{xy} = \sqrt{x} \cdot \sqrt{y} = 5 \cdot 3 = 15 \]
Therefore, \( \sqrt{xy} = 15 \).

(7) If \( a^4 = 16 \), what is the value of \( a^2 \)?

• (a) 2
• (b) 4
• (c) 8
• (d) 16

Answer

Correct Answer: (b) 4
Given that \( a^4 = 16 \), we can find \( a^2 \) by taking the square root of both sides:
\[ \sqrt{a^4} = \sqrt{16} \]
\[ a^2 = 4 \]
Therefore, \( a^2 = 4 \).

(8) Simplify \( \sqrt{18} \) as a surd.

• (a) \( 3\sqrt{2} \)
• (b) \( 6\sqrt{3} \)
• (c) \( 3\sqrt{6} \)
• (d) \( 2\sqrt{9} \)

Answer

Correct Answer: (c) \( 3\sqrt{6} \)
To simplify \( \sqrt{18} \) as a surd, we can factor 18 as \( 3 \cdot 6 \):
\[ \sqrt{18} = \sqrt{3 \cdot 6} = \sqrt{3} \cdot \sqrt{6} = 3\sqrt{6} \]
Therefore, \( \sqrt{18} \) as a surd is \( 3\sqrt{6} \).

(9) If \( \frac{a^3}{b^2} = 16 \), what is the value of \( \frac{a}{b} \)?

• (a) 2
• (b) 4
• (c) 8
• (d) 64

Answer

Correct Answer: (d) 64
Given that \( \frac{a^3}{b^2} = 16 \), we can find \( \frac{a}{b} \) by taking the cube root of both sides:
\[ \sqrt[3]{\frac{a^3}{b^2}} = \sqrt[3]{16} \]
\[ \frac{a}{b} = 2 \cdot 2 \cdot 2 = 8 \]
Therefore, \( \frac{a}{b} = 8 \).

(10) If \( \sqrt[3]{x} = 4 \) and \( \sqrt[4]{y} = 2 \), what is the value of \( \sqrt[12]{xy} \)?

• (a) 1
• (b) 2
• (c) 4
• (d) 8

Answer

Correct Answer: (c) 4
Given that \( \sqrt[3]{x} = 4 \) and \( \sqrt[4]{y} = 2 \), we can find \( \sqrt[12]{xy} \) as follows:
\[ \sqrt[12]{xy} = \sqrt[3 \cdot 4]{x \cdot y} = \sqrt[12]{4x} \]
Now, substitute the value of \( \sqrt[3]{x} \) into the expression:
\[ \sqrt[12]{4x} = \sqrt[12]{4 \cdot 4^3} = \sqrt[12]{4^4} = 4 \]
Therefore, \( \sqrt[12]{xy} = 4 \).