Surds and Indices

(31) If \( \sqrt[3]{a} = 3 \) and \( \sqrt[4]{b} = 2 \), what is the value of \( \sqrt[12]{ab} \)?

• (a) \( \sqrt[12]{6} \)
• (b) \( \sqrt[12]{18} \)
• (c) \( \sqrt[12]{24} \)
• (d) \( \sqrt[12]{36} \)

Answer

Correct Answer: (b) \( \sqrt[12]{18} \)
Given that \( \sqrt[3]{a} = 3 \) and \( \sqrt[4]{b} = 2 \), we can find \( \sqrt[12]{ab} \) as follows:
\[ \sqrt[3]{a} = 3 \implies a = 3^3 = 27 \]
\[ \sqrt[4]{b} = 2 \implies b = 2^4 = 16 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[12]{ab} = \sqrt[12]{27 \cdot 16} = \sqrt[12]{432} = \sqrt[12]{2^4 \cdot 3^3} = 2\sqrt[12]{18} \]
Therefore, \( \sqrt[12]{ab} = 2\sqrt[12]{18} \).

(32) Express \( 64^{\frac{3}{2}} \) as a surd.

• (a) \( 4\sqrt{6} \)
• (b) \( 32\sqrt{2} \)
• (c) \( 16\sqrt{3} \)
• (d) \( 128\sqrt{2} \)

Answer

Correct Answer: (b) \( 32\sqrt{2} \)
To express \( 64^{\frac{3}{2}} \) as a surd, we can first find the square root of 64 and then cube the result:
\[ 64^{\frac{3}{2}} = \left(64^{\frac{1}{2}}\right)^3 = \left(\sqrt{64}\right)^3 = \left(\sqrt{8^2}\right)^3 = (8)^3 = 512 \]
Therefore, \( 64^{\frac{3}{2}} = 32\sqrt{2} \).

(33) If \( \sqrt{x} = 12 \) and \( \sqrt{y} = 9 \), what is the value of \( \sqrt{xy} \)?

• (a) \( 105 \)
• (b) \( 108 \)
• (c) \( 99 \)
• (d) \( 96 \)

Answer

Correct Answer: (b) \( 108 \)
Given that \( \sqrt{x} = 12 \) and \( \sqrt{y} = 9 \), we can find \( \sqrt{xy} \) as follows:
\[ \sqrt{x} = 12 \implies x = 12^2 = 144 \]
\[ \sqrt{y} = 9 \implies y = 9^2 = 81 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt{xy} = \sqrt{144 \cdot 81} = \sqrt{11664} = 108 \]
Therefore, \( \sqrt{xy} = 108 \).

(34) If \( \sqrt[3]{x} = 6 \) and \( \sqrt{y} = 8 \), what is the value of \( \sqrt[6]{xy} \)?

• (a) \( 4\sqrt{2} \)
• (b) \( 12\sqrt{2} \)
• (c) \( 2\sqrt{2} \)
• (d) \( 16\sqrt{2} \)

Answer

Correct Answer: (c) \( 2\sqrt{2} \)
Given that \( \sqrt[3]{x} = 6 \) and \( \sqrt{y} = 8 \), we can find \( \sqrt[6]{xy} \) as follows:
\[ \sqrt[3]{x} = 6 \implies x = 6^3 = 216 \]
\[ \sqrt{y} = 8 \implies y = 8^2 = 64 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[6]{xy} = \sqrt[6]{216 \cdot 64} = \sqrt[6]{13824} = \sqrt[6]{2^7 \cdot 3^3} = 2\sqrt[6]{2} \]
Therefore, \( \sqrt[6]{xy} = 2\sqrt[6]{2} \).

(35) Express \( 81^{\frac{2}{3}} \) as a surd.

• (a) \( 4\sqrt{3} \)
• (b) \( 27\sqrt{2} \)
• (c) \( 16\sqrt{3} \)
• (d) \( 9\sqrt{3} \)

Answer

Correct Answer: (d) \( 9\sqrt{3} \)
To express \( 81^{\frac{2}{3}} \) as a surd, we can first find the cube root of 81 and then square the result:
\[ 81^{\frac{2}{3}} = \left(81^{\frac{1}{3}}\right)^2 = \left(\sqrt[3]{81}\right)^2 = \left(\sqrt[3]{3^4}\right)^2 = (3)^2 = 9 \]
Therefore, \( 81^{\frac{2}{3}} = 9\sqrt{3} \).