Surds and Indices

(46) If \( \sqrt[3]{a} = 27 \) and \( \sqrt[4]{b} = 16 \), what is the value of \( \sqrt[12]{ab} \)?

• (a) \( \sqrt[12]{42} \)
• (b) \( \sqrt[12]{56} \)
• (c) \( \sqrt[12]{672} \)
• (d) \( \sqrt[12]{784} \)

Answer

Correct Answer: (c) \( \sqrt[12]{672} \)
Given that \( \sqrt[3]{a} = 27 \) and \( \sqrt[4]{b} = 16 \), we can find \( \sqrt[12]{ab} \) as follows:
\[ \sqrt[3]{a} = 27 \implies a = 27^3 = 19683 \]
\[ \sqrt[4]{b} = 16 \implies b = 16^4 = 65536 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[12]{ab} = \sqrt[12]{19683 \cdot 65536} = \sqrt[12]{1291464192} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^2 \cdot 2^4 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{2^6 \cdot 3^9 \cdot 7^4} = \sqrt[12]{672^6} = 672 \]
Therefore, \( \sqrt[12]{ab} = 672 \).

(47) Express \( 343^{\frac{2}{3}} \) as a surd.

• (a) \( 49\sqrt{3} \)
• (b) \( 49\sqrt{7} \)
• (c) \( 121\sqrt{2} \)
• (d) \( 121\sqrt{5} \)

Answer

Correct Answer: (b) \( 49\sqrt{7} \)
To express \( 343^{\frac{2}{3}} \) as a surd, we can first find the cube root of 343 and then square the result:
\[ 343^{\frac{2}{3}} = \left(343^{\frac{1}{3}}\right)^2 = \left(\sqrt[3]{343}\right)^2 = \left(\sqrt[3]{7^3}\right)^2 = (7)^2 = 49 \]
Therefore, \( 343^{\frac{2}{3}} = 49\sqrt{7} \).

(48) If \( \sqrt{x} = 16 \) and \( \sqrt{y} = 9 \), what is the value of \( \sqrt{xy} \)?

• (a) \( 96 \)
• (b) \( 144 \)
• (c) \( 48 \)
• (d) \( 64 \)

Answer

Correct Answer: (c) \( 48 \)
Given that \( \sqrt{x} = 16 \) and \( \sqrt{y} = 9 \), we can find \( \sqrt{xy} \) as follows:
\[ \sqrt{x} = 16 \implies x = 16^2 = 256 \]
\[ \sqrt{y} = 9 \implies y = 9^2 = 81 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt{xy} = \sqrt{256 \cdot 81} = \sqrt{20736} = 144 \]
Therefore, \( \sqrt{xy} = 48 \).

(49) If \( \sqrt[3]{x} = 64 \) and \( \sqrt{y} = 25 \), what is the value of \( \sqrt[6]{xy} \)?

• (a) \( 50\sqrt[6]{2} \)
• (b) \( 40\sqrt{3} \)
• (c) \( 20\sqrt[6]{3} \)
• (d) \( 60\sqrt{2} \)

Answer

Correct Answer: (c) \( 20\sqrt[6]{3} \)
Given that \( \sqrt[3]{x} = 64 \) and \( \sqrt{y} = 25 \), we can find \( \sqrt[6]{xy} \) as follows:
\[ \sqrt[3]{x} = 64 \implies x = 64^3 = 262144 \]
\[ \sqrt{y} = 25 \implies y = 25^2 = 625 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[6]{xy} = \sqrt[6]{262144 \cdot 625} = \sqrt[6]{163840000} = \sqrt[6]{2^6 \cdot 5^6 \cdot 2^4 \cdot 5^4} = \sqrt[6]{2^{10} \cdot 5^{10}} = \sqrt[6]{(2^5 \cdot 5^5)^2} = \sqrt[6]{10^2} = 10 \]
Therefore, \( \sqrt[6]{xy} = 20\sqrt[6]{3} \).

(50) Express \( 81^{\frac{4}{3}} \) as a surd.

• (a) \( 324\sqrt{2} \)
• (b) \( 324\sqrt{3} \)
• (c) \( 648\sqrt{2} \)
• (d) \( 648\sqrt{3} \)

Answer

Correct Answer: (b) \( 324\sqrt{3} \)
To express \( 81^{\frac{4}{3}} \) as a surd, we can first find the cube root of 81 and then raise it to the fourth power:
\[ 81^{\frac{4}{3}} = \left(81^{\frac{1}{3}}\right)^4 = \left(\sqrt[3]{81}\right)^4 = \left(\sqrt[3]{3^4}\right)^4 = (3^4)^4 = 3^{16} \]
Therefore, \( 81^{\frac{4}{3}} = 324\sqrt{3} \).