Surds & Indices Questions and Answers

Answer: B

\((10)^{150}\div (10)^{146}=\frac{10^{150}}{10^{146}}\)

\(=10^{150-146}\)

\(=10^{4}\)

\(=10000\)

Answer: A

\(3^x - 3^{x-1} = 18 \)

\(\Rightarrow 3^{x-1} \times (3-1) = 18 \)

\(\Rightarrow 3^{x-1} = 9 = 3^2 \)

\(\Rightarrow x-1 = 2 \)

\(\Rightarrow x = 3\).

Answer: B

\(3^{x-y} =27 = 3^{3} \Leftrightarrow x-y=3\) ...(i)

\(3^{x+y}=243=3^{5} \Leftrightarrow x+y=5\) ...(ii)

On solving (i) and (ii), we get \(x=4\).

Answer: C

Exp: \(((3^{y})^{\sqrt{}2})^{2} = 729\)

\((3^y)^2 = 3^4 (\sqrt2^2 = (2^{\frac{1}{2}}) = 2)\)

equating powers of 2 on both sides, 

\(y^2 = 4\)

\(\Rightarrow y = \pm2\)

Answer: A

Exp: Let \(\sqrt{[200\sqrt{[200\sqrt{[200....... \infty ]}]}]} = x\).

Hence \(\sqrt{200x} = x\)

Squaring both sides \(200x = x^2\)

\(\Rightarrow x(x-200) = 0\)

\(\Rightarrow x = 0 \) or \(x-200 = 0\) i.e. \(x = 200\)

As \(x\) cannot be 0, \(x = 200\)

Answer: A

Correct answer: 10

Surds multiply under a single root sign.

\[\sqrt{5} \times \sqrt{20} = \sqrt{100} = 10\]

Answering 100 stops at the product inside the root without taking the square root.

Answer: D

Let \((17)^{3.5}\times (17)^{x}=17^{8}\)

Then, \((17)^{3.5+x}=17^{8}\)

\(\therefore 3.5+x=8 \)

\(\Rightarrow x=(8-3.5) \)

\(\Rightarrow x=4.5\)

Answer: D

Let \(a^x = b^y = c^z = k\). 

Then, \(a = k^{\frac{1}{x}}, b = k^{\frac{1}{y}}, c = k^{\frac{1}{z}}\)

Therefore, \(b^2 = ac \)

\(\Rightarrow (k^{\frac{1}{y}})^2 = k^{\frac{1}{x}} \times k^{\frac{1}{z}} \)

\(\Rightarrow k^{\frac{2}{y}} = k^{(\frac{1}{x} + \frac{1}{z})}\)

Therefore, \(\frac{2}{y} = \frac{(x+z)}{xz} \)

\(\Rightarrow \frac{y}{2} = \frac{xz}{(x+z)} \)

\(\Rightarrow y = \frac{2xz}{(x + z)}\)

Answer: D

Exp: \(4^{4m+2} = (2^3)^{6m-4} \)

\(\Rightarrow 4^{4m+2} = 2^{18m+12}\)

Equating powers of 2 both sides,

\(4m + 2 = 18m -12 \)

\(\Rightarrow 14=14m \)

\(\Rightarrow m=1\)

Answer: A

\(5^{a}=3125 \Leftrightarrow 5^{a}=5^{5} \)

\(\Rightarrow a = 5 \)

\(\therefore 5^{(a-3)}=5^{(5-3)}=5^{2}=25\)