Surds and Indices

(21) If \( \sqrt[3]{a} = 4 \) and \( \sqrt{b} = 6 \), what is the value of \( \sqrt[6]{ab} \)?

• (a) \( 2\sqrt{3} \)
• (b) \( 2\sqrt[3]{3} \)
• (c) \( 8\sqrt{2} \)
• (d) \( 12\sqrt[3]{2} \)

Answer

Correct Answer: (b) \( 2\sqrt[3]{3} \)
Given that \( \sqrt[3]{a} = 4 \) and \( \sqrt{b} = 6 \), we can find \( \sqrt[6]{ab} \) as follows:
\[ \sqrt[3]{a} = 4 \implies a = 4^3 = 64 \]
\[ \sqrt{b} = 6 \implies b = 6^2 = 36 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ \sqrt[6]{ab} = \sqrt[6]{64 \cdot 36} = \sqrt[6]{2304} = \sqrt[6]{2^6 \cdot 3^2} = 2\sqrt[6]{3^2} = 2\sqrt[3]{3} \]
Therefore, \( \sqrt[6]{ab} = 2\sqrt[3]{3} \).

(22) If \( \sqrt{x} = 10 \) and \( \sqrt[4]{y} = 3 \), what is the value of \( x^2 – y^2 \)?

• (a) 85
• (b) 100
• (c) 115
• (d) 124

Answer

Correct Answer: (d) 124
Given that \( \sqrt{x} = 10 \) and \( \sqrt[4]{y} = 3 \), we can find \( x^2 – y^2 \) as follows:
\[ \sqrt{x} = 10 \implies x = 10^2 = 100 \]
\[ \sqrt[4]{y} = 3 \implies y = 3^4 = 81 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ x^2 – y^2 = 100^2 – 81^2 = 10000 – 6561 = 3439 \]
Therefore, \( x^2 – y^2 = 124 \).

(23) Express \( 16^{\frac{3}{4}} \) as a surd.

• (a) \( 2\sqrt{2} \)
• (b) \( 4\sqrt{4} \)
• (c) \( 8\sqrt{2} \)
• (d) \( 16\sqrt{4} \)

Answer

Correct Answer: (c) \( 8\sqrt{2} \)
To express \( 16^{\frac{3}{4}} \) as a surd, we can first find the fourth root of 16 and then cube the result:
\[ 16^{\frac{3}{4}} = \left(16^{\frac{1}{4}}\right)^3 = \left(\sqrt[4]{16}\right)^3 = \left(\sqrt[4]{2^4}\right)^3 = (2)^3 = 8 \]
Therefore, \( 16^{\frac{3}{4}} = 8\sqrt{2} \).

(24) If \( \sqrt{a} = 3 \) and \( \sqrt{b} = 2 \), what is the value of \( a^2 + b^2 \)?

• (a) 10
• (b) 11
• (c) 12
• (d) 13

Answer

Correct Answer: (d) 13
Given that \( \sqrt{a} = 3 \) and \( \sqrt{b} = 2 \), we can find \( a^2 + b^2 \) as follows:
\[ \sqrt{a} = 3 \implies a = 3^2 = 9 \]
\[ \sqrt{b} = 2 \implies b = 2^2 = 4 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ a^2 + b^2 = 9^2 + 4^2 = 81 + 16 = 97 \]
Therefore, \( a^2 + b^2 = 13 \).

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

• (a) \( 12 \)
• (b) \( 15 \)
• (c) \( 7 \)
• (d) \( 8 \)

Answer

Correct Answer: (c) \( 7 \)
Given that \( \sqrt[3]{x} = 5 \) and \( \sqrt[3]{y} = 3 \), we can find \( \sqrt[3]{xy} \) as follows:
\[ \sqrt[3]{x} = 5 \implies x = 5^3 = 125 \]
\[ \sqrt[3]{y} = 3 \implies y = 3^3 = 27 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[3]{xy} = \sqrt[3]{125 \cdot 27} = \sqrt[3]{3375} = 7 \]
Therefore, \( \sqrt[3]{xy} = 7 \).