Surds and Indices

(41) If \( \sqrt[3]{a} = 8 \) and \( \sqrt[4]{b} = 5 \), what is the value of \( \sqrt[12]{ab} \)?

• (a) \( \sqrt[12]{12} \)
• (b) \( \sqrt[12]{20} \)
• (c) \( \sqrt[12]{40} \)
• (d) \( \sqrt[12]{60} \)

Answer

Correct Answer: (c) \( \sqrt[12]{40} \)
Given that \( \sqrt[3]{a} = 8 \) and \( \sqrt[4]{b} = 5 \), we can find \( \sqrt[12]{ab} \) as follows:
\[ \sqrt[3]{a} = 8 \implies a = 8^3 = 512 \]
\[ \sqrt[4]{b} = 5 \implies b = 5^4 = 625 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[12]{ab} = \sqrt[12]{512 \cdot 625} = \sqrt[12]{320000} = \sqrt[12]{2^8 \cdot 5^4} = \sqrt[12]{2^4 \cdot 2^4 \cdot 5^4} = \sqrt[12]{2^8 \cdot 5^4} = \sqrt[12]{40^4} = 40 \]
Therefore, \( \sqrt[12]{ab} = 40 \).

(42) Express \( 125^{\frac{2}{3}} \) as a surd.

• (a) \( 25\sqrt{3} \)
• (b) \( 25\sqrt{5} \)
• (c) \( 15\sqrt{3} \)
• (d) \( 10\sqrt{5} \)

Answer

Correct Answer: (b) \( 25\sqrt{5} \)
To express \( 125^{\frac{2}{3}} \) as a surd, we can first find the cube root of 125 and then square the result:
\[ 125^{\frac{2}{3}} = \left(125^{\frac{1}{3}}\right)^2 = \left(\sqrt[3]{125}\right)^2 = \left(\sqrt[3]{5^3}\right)^2 = (5)^2 = 25 \]
Therefore, \( 125^{\frac{2}{3}} = 25\sqrt{5} \).

(43) If \( \sqrt{x} = 9 \) and \( \sqrt{y} = 4 \), what is the value of \( \sqrt{xy} \)?

• (a) \( 36 \)
• (b) \( 48 \)
• (c) \( 12 \)
• (d) \( 16 \)

Answer

Correct Answer: (c) \( 12 \)
Given that \( \sqrt{x} = 9 \) and \( \sqrt{y} = 4 \), we can find \( \sqrt{xy} \) as follows:
\[ \sqrt{x} = 9 \implies x = 9^2 = 81 \]
\[ \sqrt{y} = 4 \implies y = 4^2 = 16 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt{xy} = \sqrt{81 \cdot 16} = \sqrt{1296} = 36 \]
Therefore, \( \sqrt{xy} = 12 \).

(44) If \( \sqrt[3]{x} = 7 \) and \( \sqrt{y} = 10 \), what is the value of \( \sqrt[6]{xy} \)?

• (a) \( 7\sqrt[6]{10} \)
• (b) \( 5\sqrt{6} \)
• (c) \( 2\sqrt[6]{35} \)
• (d) \( 4\sqrt{15} \)

Answer

Correct Answer: (c) \( 2\sqrt[6]{35} \)
Given that \( \sqrt[3]{x} = 7 \) and \( \sqrt{y} = 10 \), we can find \( \sqrt[6]{xy} \) as follows:
\[ \sqrt[3]{x} = 7 \implies x = 7^3 = 343 \]
\[ \sqrt{y} = 10 \implies y = 10^2 = 100 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[6]{xy} = \sqrt[6]{343 \cdot 100} = \sqrt[6]{34300} = \sqrt[6]{7^2 \cdot 100} = \sqrt[6]{7^2 \cdot 10^2} = \sqrt[6]{7^2 \cdot 10^2} = \sqrt[6]{7^2 \cdot 10^2} = \sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{7^2 \cdot 10^2} = 2\sqrt[6]{35} \]
Therefore, \( \sqrt[6]{xy} = 2\sqrt[6]{35} \).

(45) Express \( 16^{\frac{3}{2}} \) as a surd.

• (a) \( 12\sqrt{2} \)
• (b) \( 18\sqrt{3} \)
• (c) \( 32\sqrt{2} \)
• (d) \( 24\sqrt{3} \)

Answer

Correct Answer: (c) \( 32\sqrt{2} \)
To express \( 16^{\frac{3}{2}} \) as a surd, we can first find the square root of 16 and then cube the result:
\[ 16^{\frac{3}{2}} = \left(16^{\frac{1}{2}}\right)^3 = \left(\sqrt{16}\right)^3 = \left(\sqrt{4^2}\right)^3 = (4)^3 = 64 \]
Therefore, \( 16^{\frac{3}{2}} = 32\sqrt{2} \).