The value of \((\frac{32}{243})^{-\frac{4}{5}}\) is:
Answer: D
\((\frac{32}{243})^{-\frac{4}{5}} \)
\(= (\frac{243}{32})^{\frac{4}{5}} \)
\(= [(\frac{3}{2})^5]^{\frac{4}{5}} \)
\(= \frac{81}{16}\)
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\((\frac{1}{216})^{-\frac{2}{3}}\div(\frac{1}{27})^{-\frac{4}{3}}=?\)
Answer: C
\((\frac{1}{216})^{-\frac{2}{3}}\div(\frac{1}{27})^{-\frac{4}{3}} \)
\(= 216^{\frac{2}{3}}\div27^{\frac{4}{3}} \)
\(= (63)^{\frac{2}{3}}\div(33)^{\frac{4}{3}} \)
\(= \frac{4}{9}\)
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\[ \frac{2^{n+4} - 2 \cdot 2^{n}}{2 \cdot 2^{n+3}} + 2^{-3} \ \text{ is equal to:} \]
Answer: D
Factor \( 2^{n} \) out of the fraction.
\[ \frac{2^{n+4} - 2 \cdot 2^{n}}{2 \cdot 2^{n+3}} = \frac{2^{n}(16 - 2)}{2^{n}(16)} = \frac{14}{16} = \frac{7}{8} \]
\[ \frac{7}{8} + 2^{-3} = \frac{7}{8} + \frac{1}{8} = 1 \]
The \( 2^{n} \) cancels entirely, so the answer does not depend on \( n \).
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If \(5\sqrt{5}\times5^3\div5^{-\frac{3}{2}}=5^{a+2}\), the value of \(a\) is:
Answer: A
\(5^{\frac{3}{2}}\times5^3\div5^{3\frac{3}{2}}=5^{a+2} \)
\(\Rightarrow 5^{\frac{3}{2}}+3+\frac{3}{2}=5^{a+2} \)
\(\Rightarrow \frac{3}{2}+3+\frac{3}{2}=a+2 \)
\(\Rightarrow a+2=6 \)
\(\therefore a=4\)
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If \(\sqrt{2n}=64\), then the value of \(n\) is:
Answer: D
\(\sqrt{2n}=64 \)
\(\Rightarrow 2^{\frac{n}{2}}=64=2^6 \)
\(\Rightarrow \frac{n}{2}=6 \)
\(\therefore n=12\)
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The simplified form of \((x^{\frac{7}{2}}\div x^{\frac{5}{2}})\times \sqrt{y3}\div \sqrt{y}\) is:
Answer: D
\((x^{\frac{7}{2}}\div x^{\frac{5}{2}})\times \sqrt{y3}\div \sqrt{y} \)
\(= x^{(\frac{7}{2}-\frac{5}{2})}\times y^{(\frac{3}{2}-\frac{1}{2})} \)
\(= xy\)
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If \(m\) and \(n\) are whole numbers such that \(m^n = 169\), then the value of \((m-1)^{n+1}\) is:
Answer: D
\( 169 = 13^{2} \), so \( m = 13 \) and \( n = 2 \).
\[ (m-1)^{n+1} = 12^{3} = 1728 \]
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The simplified form of \(\frac{x^{\frac{9}{2}}\times \sqrt{y^5}}{x^{\frac{7}{2}}\times \sqrt{y^3}}\) is:
Answer: B
Exp: \(\frac{x^{\frac{9}{2}}\times \sqrt{y^5}}{x^{\frac{7}{2}}\times \sqrt{y^3}}\) is:
\(= x^{\frac{9}{2}-\frac{5}{2}}\times y^{\frac{7}{2}-\frac{3}{2}} \)
\(= x^2 . y^2\)
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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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If \(x\) is an integer, find the minimum value of \(x\) such that \(0.00001154111\times 10^x\) exceeds 1000.
Answer: A
Exp: Considering from the left if the decimal point is shifted by 8 places to the right, the number
becomes 1154.111. Therefore, \(0.00001154111\times 10^x\) exceeds 1000 when \(x\) has a minimum value of 8.
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