Surds and Indices

(16) If \( \sqrt[4]{a} = 2 \) and \( \sqrt{b} = 5 \), what is the value of \( \sqrt[8]{ab} \)?

• (a) \( 5 \)
• (b) \( 2.5 \)
• (c) \( 10 \)
• (d) \( 12.5 \)

Answer

Correct Answer: (b) \( 2.5 \)
Given that \( \sqrt[4]{a} = 2 \) and \( \sqrt{b} = 5 \), we can find \( \sqrt[8]{ab} \) as follows:
\[ \sqrt[4]{a} = 2 \implies a = 2^4 = 16 \]
\[ \sqrt{b} = 5 \implies b = 5^2 = 25 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[8]{ab} = \sqrt[8]{16 \cdot 25} = \sqrt[8]{400} = \sqrt[8]{4^2 \cdot 25} = 4\sqrt[8]{25} = 2.5 \]
Therefore, \( \sqrt[8]{ab} = 2.5 \).

(17) If \( \sqrt{a} = 7 \) and \( \sqrt{b} = 9 \), what is the value of \( \frac{a}{b} \)?

• (a) \( \frac{7}{9} \)
• (b) \( \frac{49}{81} \)
• (c) \( \frac{9}{7} \)
• (d) \( \frac{81}{49} \)

Answer

Correct Answer: (b) \( \frac{49}{81} \)
Given that \( \sqrt{a} = 7 \) and \( \sqrt{b} = 9 \), we can find \( \frac{a}{b} \) as follows:
\[ \sqrt{a} = 7 \implies a = 7^2 = 49 \]
\[ \sqrt{b} = 9 \implies b = 9^2 = 81 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \frac{a}{b} = \frac{49}{81} \]
Therefore, \( \frac{a}{b} = \frac{49}{81} \).

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

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

Answer

Correct Answer: (a) \( 2\sqrt[6]{3} \)
Given that \( \sqrt[3]{x} = 2 \) and \( \sqrt[6]{y} = 3 \), we can find \( \sqrt[6]{xy} \) as follows:
\[ \sqrt[3]{x} = 2 \implies x = 2^3 = 8 \]
\[ \sqrt[6]{y} = 3 \implies y = 3^6 = 729 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[6]{xy} = \sqrt[6]{8 \cdot 729} = \sqrt[6]{5832} = \sqrt[6]{2^3 \cdot 3^6} = 2\sqrt[6]{3} \]
Therefore, \( \sqrt[6]{xy} = 2\sqrt[6]{3} \).

(19) If \( \sqrt[5]{x} = 4 \) and \( \sqrt[10]{y} = 2 \), what is the value of \( \sqrt[10]{xy} \)?

• (a) \( 2\sqrt[10]{4} \)
• (b) \( 4\sqrt[10]{2} \)
• (c) \( 2\sqrt[10]{8} \)
• (d) \( 8\sqrt[10]{2} \)

Answer

Correct Answer: (c) \( 2\sqrt[10]{8} \)
Given that \( \sqrt[5]{x} = 4 \) and \( \sqrt[10]{y} = 2 \), we can find \( \sqrt[10]{xy} \) as follows:
\[ \sqrt[5]{x} = 4 \implies x = 4^5 = 1024 \]
\[ \sqrt[10]{y} = 2 \implies y = 2^{10} = 1024 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[10]{xy} = \sqrt[10]{1024 \cdot 1024} = \sqrt[10]{1048576} = \sqrt[10]{2^{20}} = 2\sqrt[10]{2^2 \cdot 2^{18}} = 2\sqrt[10]{2^2 \cdot 4^{9}} = 2\sqrt[10]{8^9} = 2\sqrt[10]{8} \]
Therefore, \( \sqrt[10]{xy} = 2\sqrt[10]{8} \).

(20) If \( \sqrt{x} = 7 \) and \( \sqrt{y} = 5 \), what is the value of \( \sqrt[4]{xy} \)?

• (a) \( \sqrt{6} \)
• (b) \( \sqrt{35} \)
• (c) \( \sqrt{12} \)
• (d) \( \sqrt{25} \)

Answer

Correct Answer: (b) \( \sqrt{35} \)
Given that \( \sqrt{x} = 7 \) and \( \sqrt{y} = 5 \), we can find \( \sqrt[4]{xy} \) as follows:
\[ \sqrt{x} = 7 \implies x = 7^2 = 49 \]
\[ \sqrt{y} = 5 \implies y = 5^2 = 25 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[4]{xy} = \sqrt[4]{49 \cdot 25} = \sqrt[4]{1225} = \sqrt{35} \]
Therefore, \( \sqrt[4]{xy} = \sqrt{35} \).