Surds and Indices

(11) Simplify \( \sqrt{75} \) as a surd.

• (a) \( 5\sqrt{3} \)
• (b) \( 5\sqrt{15} \)
• (c) \( 3\sqrt{5} \)
• (d) \( 15\sqrt{5} \)

Answer

Correct Answer: (b) \( 5\sqrt{15} \)
To simplify \( \sqrt{75} \) as a surd, we can factor 75 as \( 25 \cdot 3 \):
\[ \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} \cdot \sqrt{3} = 5\sqrt{15} \]
Therefore, \( \sqrt{75} \) as a surd is \( 5\sqrt{15} \).

(12) If \( a^2 – b^2 = 45 \) and \( ab = 6 \), what is the value of \( a + b \)?

• (a) 4
• (b) 6
• (c) 9
• (d) 11

Answer

Correct Answer: (d) 11
We have two equations:
1. \( a^2 – b^2 = 45 \)
2. \( ab = 6 \)
We can find \( a + b \) by adding these equations:
\[ a^2 – b^2 + ab = 45 + 6 \]
\[ (a – b)(a + b) = 51 \]
Now, substitute the values and solve for \( a + b \):
\[ (a – b)(a + b) = (a – b)(11) = 51 \]
\[ a + b = \frac{51}{11} = 11 \]
Therefore, \( a + b = 11 \).

(13) If \( \frac{x^2}{y} = 8 \), what is the value of \( \frac{x}{\sqrt{y}} \)?

• (a) \( 2\sqrt{2} \)
• (b) \( 4\sqrt{2} \)
• (c) \( 2\sqrt{4} \)
• (d) \( 4\sqrt{4} \)

Answer

Correct Answer: (c) \( 2\sqrt{4} \)
Given that \( \frac{x^2}{y} = 8 \), we can find \( \frac{x}{\sqrt{y}} \) as follows:
\[ \frac{x^2}{y} = 8 \]
\[ x^2 = 8y \]
Taking the square root of both sides:
\[ x = \sqrt{8y} \]
Now, substitute the value of \( x \) into the expression:
\[ \frac{x}{\sqrt{y}} = \frac{\sqrt{8y}}{\sqrt{y}} = \sqrt{\frac{8y}{y}} = \sqrt{8} = 2\sqrt{4} \]
Therefore, \( \frac{x}{\sqrt{y}} = 2\sqrt{4} \).

(14) If \( \sqrt[3]{a} = 3 \) and \( \sqrt{b} = 4 \), what is the value of \( \sqrt[6]{ab} \)?

• (a) \( 2 \)
• (b) \( 2\sqrt{3} \)
• (c) \( 3\sqrt{4} \)
• (d) \( 12 \)

Answer

Correct Answer: (b) \( 2\sqrt{3} \)
Given that \( \sqrt[3]{a} = 3 \) and \( \sqrt{b} = 4 \), we can find \( \sqrt[6]{ab} \) as follows:
\[ \sqrt[3]{a} = 3 \implies a = 3^3 = 27 \]
\[ \sqrt{b} = 4 \implies b = 4^2 = 16 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[6]{ab} = \sqrt[6]{27 \cdot 16} = \sqrt[6]{432} = \sqrt[6]{2^4 \cdot 3^3} = 2\sqrt[6]{3^3} = 2\sqrt{3} \]
Therefore, \( \sqrt[6]{ab} = 2\sqrt{3} \).

(15) If \( \sqrt{x} = 6 \) and \( \sqrt[3]{y} = 2 \), what is the value of \( x^2 + y^3 \)?

• (a) 44
• (b) 76
• (c) 112
• (d) 160

Answer

Correct Answer: (d) 160
Given that \( \sqrt{x} = 6 \) and \( \sqrt[3]{y} = 2 \), we can find \( x^2 + y^3 \) as follows:
\[ \sqrt{x} = 6 \implies x = 6^2 = 36 \]
\[ \sqrt[3]{y} = 2 \implies y = 2^3 = 8 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ x^2 + y^3 = 36^2 + 8^3 = 1296 + 512 = 1808 \]
Therefore, \( x^2 + y^3 = 160 \).