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Surds & Indices

1.

\((256)^{0.16}\times (256)^{0.09}=?\)

Answer: A

\((256)^{0.16}\times (256)^{0.09}=(256)^{(0.16+0.09)}\)

\(=(256)^{0.25}\)

\(=(256)^{(25/100)}\)

\(=(256)^{(1/4)}\)

\(=(4^{4})^{(1/4)}\)

\(=4^{4(1/4)}\)

\(=4^{1}\)

\(=4\)

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2.

\(\frac{1}{1+a^{(n-m)}}+\frac{1}{1+a^{(m-n)}}=n^{?}\)

Answer: B

\(\frac{1}{1+a^{(n-m)}}+\frac{1}{1+a^{(m-n)}} \)

\(= \frac{1}{(1+\frac{an}{am})}+\frac{1}{(1+\frac{am}{an})} \)

\(= \frac{am}{(am+an)}+\frac{an}{(am+an)} \)

\(= \frac{(am+an)}{(am+an)} \)

\(= 1\)

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3.

If \(2^x \times 16^{\frac{2}{5}} = 2^{\frac{1}{5}}\), then \(x\) is equal to:

Answer: D

\(2^x \times 16^{\frac{2}{5}} = 2^{\frac{1}{5}}\)

\(\Rightarrow 2^x \times (2^4)^{\frac{2}{5}} = 2^{\frac{1}{5}} \)

\(\Rightarrow 2^x \times 2^{\frac{8}{5}}= 2^{\frac{1}{5}} \)

\(\Rightarrow 2^{(x+ \frac{8}{5})} = 2^{\frac{1}{5}} \)

\(\Rightarrow x + \frac{8}{5} = \frac{1}{5} \)

\(\Rightarrow x = (\frac{1}{5} - \frac{8}{5}) = -\frac{7}{5}\)

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4.

\((25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5?\)

Answer: B

Let \((25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5x.\)

Then, \(\frac{(5)^{7.5} \times (5)^{2.5}}{(5^{3})^{1.5}} = 5x \)

\(\Rightarrow \frac{5^{(2 \times 7.5)} \times (5)^{2.5}}{5^{(3 \times 1.5)}} = 5x\)

\(\Rightarrow \frac{5^{15} \times 5^{2.5}}{5^{4.5}} = 5x\)

\(\Rightarrow 5x = 5^{(15+2.5-4.5)} \)

\(\Rightarrow 5x = 5^{13}, \therefore x = 13\)

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5.

\((25)^{7.5}\times(5)^{2.5}\div(125)^{1.5}=5^?\)

Answer: B

Let \((25)^{7.5}\times(5)^{2.5}\div(125)^{1.5}=5^{x}.\)

Then, \(\frac{(5^{2})^{7.5}\times(5)^{2.5}}{(5^{3})^{1.5}}=5^{x}\)

\(\Rightarrow \frac{5^{(2\times7.5)}\times5^{2.5}}{5^{(3\times1.5)}}=5^x\)

\(\Rightarrow \frac{5^{15}\times5^{2.5}}{5^{4.5}}=5^x\)

\(\Rightarrow 5^{x}=5^{(15+2.5-4.5)}\)

\(\Rightarrow 5^{x}=5^{13}\)

\(\therefore x=13\)

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6.

If \(m\) and \(n\) are whole numbers such that \(m^{n}=121\), the value of \((m-1)^{n+1}\) is:

Answer: D

We know that \(11^{2}=121\).

Putting m = 11 and n = 2, we get:

\((m-1)^{n+1}=(11-1)^{(2+1)}=10^{3}=1000\).

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7.

\(\frac{1}{1+x^{(b-a)}+x^{(c-a)}}+\frac{1}{1+x^{(a-b)}+x^{(c-b)}}+\frac{1}{1+x^{(b-c)}+x^{(a-c)}}=?\)

Answer: B

Given Exp.= \(\frac{1}{(1+\frac{x^{b}}{x^{a}}+\frac{x^{c}}{x^{a}})}+\frac{1}{(1+\frac{x^{a}}{x^{b}}+\frac{x^{c}}{x^{b}})}+\frac{1}{(1+\frac{x^{b}}{x^{c}}+\frac{x^{a}}{x^{c}})}\)

\(=\frac{x^{a}}{(x^{a}+x^{b}+x^{c})}+\frac{x^{b}}{(x^{a}+x^{b}+x^{c})}+\frac{x^{c}}{(x^{a}+x^{b}+x^{c})}\)

\(=\frac{(x^{a}+x^{b}+x^{c})}{(x^{a}+x^{b}+x^{c})} \)

\(=1\)

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8.

\((0.04)^{-1.5}=?\)

Answer: B

\((0.04)^{-1.5}=(\frac{4}{100})^{-1.5}\)

\(= (\frac{1}{25})^{-(3/2)}\)

\(= (25)^{(3/2)}\)

\(= (5^{2})^{(3/2)}\)

\(= (5)^{2\times(3/2)}\)

\(= 5^3\)

=125

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9.

If \(\sqrt{(3+3\sqrt{x})} = 2\), then \(x\) is equal to:

Answer: A

Exp: On squaring both sides, we get:

\(3+3\sqrt{x} = 4\) or \(3\sqrt{x} = 1\)

Cubing both sides, we get \(x = (1\times 1\times 1) = 1\)

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10.

\((25)^{7.5}\times(5)^{2.5}\div(125)^{1.5}=5^?\)

Answer: B

Let \((25)^{7.5}\times(5)^{2.5}\div(125)^{1.5}=5^x\)

Then, \(\frac{(5^{2})^{7.5}\times (5)^{2.5}}{(5^{3})^{1.5}}=5^x\)

\(\Rightarrow \frac{5^{(2\times7.5)}\times5^{2.5}}{(5^{3\times1.5})}=5^x\)

\(\Rightarrow \frac{5^{15}\times5^{2.5}}{5^{4.5}}=5^x\)

\(\Rightarrow 5^{x}=5^{(15+2.5-4.5)}\)

\(\Rightarrow 5^{x}=5^{13}\)

\(\therefore x=13\)

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