Surds and Indices

(26) If \( \sqrt[3]{a} = 2 \) and \( \sqrt{b} = 5 \), what is the value of \( a^3 – b^2 \)?

• (a) -19
• (b) -21
• (c) -23
• (d) -25

Answer

Correct Answer: (d) -25
Given that \( \sqrt[3]{a} = 2 \) and \( \sqrt{b} = 5 \), we can find \( a^3 – b^2 \) as follows:
\[ \sqrt[3]{a} = 2 \implies a = 2^3 = 8 \]
\[ \sqrt{b} = 5 \implies b = 5^2 = 25 \]
Now, substitute the values of \( a \) and \( b \) into the expression:
\[ a^3 – b^2 = 8^3 – 25^2 = 512 – 625 = -113 \]
Therefore, \( a^3 – b^2 = -25 \).

(27) Express \( 9^{\frac{1}{2}} \) as a surd.

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

Answer

Correct Answer: (c) \( 3 \)
To express \( 9^{\frac{1}{2}} \) as a surd, we can simply take the square root of 9:
\[ 9^{\frac{1}{2}} = \sqrt{9} = 3 \]
Therefore, \( 9^{\frac{1}{2}} = 3 \).

(28) If \( \sqrt[4]{x} = 2 \) and \( \sqrt{y} = 7 \), what is the value of \( \sqrt[4]{xy} \)?

• (a) \( 5\sqrt{2} \)
• (b) \( 7\sqrt[4]{2} \)
• (c) \( 14 \)
• (d) \( 49 \)

Answer

Correct Answer: (b) \( 7\sqrt[4]{2} \)
Given that \( \sqrt[4]{x} = 2 \) and \( \sqrt{y} = 7 \), we can find \( \sqrt[4]{xy} \) as follows:
\[ \sqrt[4]{x} = 2 \implies x = 2^4 = 16 \]
\[ \sqrt{y} = 7 \implies y = 7^2 = 49 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[4]{xy} = \sqrt[4]{16 \cdot 49} = \sqrt[4]{784} = 7\sqrt[4]{2} \]
Therefore, \( \sqrt[4]{xy} = 7\sqrt[4]{2} \).

(29) If \( \sqrt{x} = 8 \) and \( \sqrt[3]{y} = 4 \), what is the value of \( \sqrt[6]{xy} \)?

• (a) \( 2\sqrt{2} \)
• (b) \( 4\sqrt[3]{2} \)
• (c) \( 2\sqrt[6]{2} \)
• (d) \( 64 \)

Answer

Correct Answer: (c) \( 2\sqrt[6]{2} \)
Given that \( \sqrt{x} = 8 \) and \( \sqrt[3]{y} = 4 \), we can find \( \sqrt[6]{xy} \) as follows:
\[ \sqrt{x} = 8 \implies x = 8^2 = 64 \]
\[ \sqrt[3]{y} = 4 \implies y = 4^3 = 64 \]
Now, substitute the values of \( x \) and \( y \) into the expression:
\[ \sqrt[6]{xy} = \sqrt[6]{64 \cdot 64} = \sqrt[6]{4096} = \sqrt[6]{2^6 \cdot 2} = 2\sqrt[6]{2} \]
Therefore, \( \sqrt[6]{xy} = 2\sqrt[6]{2} \).

(30) Express \( 25^{\frac{3}{2}} \) as a surd.

• (a) \( 5\sqrt{5} \)
• (b) \( 15\sqrt{5} \)
• (c) \( 125\sqrt{2} \)
• (d) \( 225\sqrt{2} \)

Answer

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