Surds and Indices

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

• (a) \( \sqrt[12]{7} \)
• (b) \( \sqrt[12]{10} \)
• (c) \( \sqrt[12]{12} \)
• (d) \( \sqrt[12]{15} \)

Answer

Correct Answer: (c) \( \sqrt[12]{12} \)
Given that \( \sqrt[3]{a} = 4 \) and \( \sqrt[4]{b} = 3 \), we can find \( \sqrt[12]{ab} \) as follows:
\[ \sqrt[3]{a} = 4 \implies a = 4^3 = 64 \]
\[ \sqrt[4]{b} = 3 \implies b = 3^4 = 81 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[12]{ab} = \sqrt[12]{64 \cdot 81} = \sqrt[12]{5184} = \sqrt[12]{2^7 \cdot 3^4} = \sqrt[12]{2^3 \cdot 2^4 \cdot 3^4} = \sqrt[12]{2^7 \cdot 3^4} = \sqrt[12]{12^4} = 12 \]
Therefore, \( \sqrt[12]{ab} = 12 \).

(37) Express \( 36^{\frac{2}{3}} \) as a surd.

• (a) \( 18\sqrt{2} \)
• (b) \( 12\sqrt{3} \)
• (c) \( 24\sqrt{2} \)
• (d) \( 9\sqrt{4} \)

Answer

Correct Answer: (b) \( 12\sqrt{3} \)
To express \( 36^{\frac{2}{3}} \) as a surd, we can first find the cube root of 36 and then square the result:
\[ 36^{\frac{2}{3}} = \left(36^{\frac{1}{3}}\right)^2 = \left(\sqrt[3]{36}\right)^2 = \left(\sqrt[3]{6^2}\right)^2 = (6)^2 = 36 \]
Therefore, \( 36^{\frac{2}{3}} = 12\sqrt{3} \).

(38) If \( \sqrt{x} = 10 \) and \( \sqrt{y} = 6 \), what is the value of \( \sqrt{xy} \)?

• (a) \( 56 \)
• (b) \( 60 \)
• (c) \( 30 \)
• (d) \( 36 \)

Answer

Correct Answer: (c) \( 30 \)
Given that \( \sqrt{x} = 10 \) and \( \sqrt{y} = 6 \), we can find \( \sqrt{xy} \) as follows:
\[ \sqrt{x} = 10 \implies x = 10^2 = 100 \]
\[ \sqrt{y} = 6 \implies y = 6^2 = 36 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt{xy} = \sqrt{100 \cdot 36} = \sqrt{3600} = 60 \]
Therefore, \( \sqrt{xy} = 30 \).

(39) If \( \sqrt[3]{x} = 5 \) and \( \sqrt{y} = 7 \), what is the value of \( \sqrt[6]{xy} \)?

• (a) \( 2\sqrt[6]{35} \)
• (b) \( 3\sqrt{5} \)
• (c) \( 2\sqrt[6]{70} \)
• (d) \( 4\sqrt{10} \)

Answer

Correct Answer: (c) \( 2\sqrt[6]{70} \)
Given that \( \sqrt[3]{x} = 5 \) and \( \sqrt{y} = 7 \), we can find \( \sqrt[6]{xy} \) as follows:
\[ \sqrt[3]{x} = 5 \implies x = 5^3 = 125 \]
\[ \sqrt{y} = 7 \implies y = 7^2 = 49 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[6]{xy} = \sqrt[6]{125 \cdot 49} = \sqrt[6]{6125} = \sqrt[6]{5^2 \cdot 7^2 \cdot 5} = \sqrt[6]{5^2 \cdot 7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{49 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{7^2 \cdot 5^{\frac{1}{2}}} = 5\sqrt[6]{70} \]
Therefore, \( \sqrt[6]{xy} = 2\sqrt[6]{70} \).

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

• (a) \( 48\sqrt{2} \)
• (b) \( 32\sqrt{3} \)
• (c) \( 16\sqrt{6} \)
• (d) \( 96\sqrt{2} \)

Answer

Correct Answer: (b) \( 32\sqrt{3} \)
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, \( 64^{\frac{3}{2}} = 32\sqrt{3} \).