If \(a^x = b, b^y = c\) and \(c^z = a\), then the value of \(xyz\) is:
Answer: A
Substitute each equation into the next.
\[ c = b^{y} = (a^{x})^{y} = a^{xy} \]
\[ a = c^{z} = (a^{xy})^{z} = a^{xyz} \]
Comparing exponents on the same base:
\[ xyz = 1 \]
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If \(7^a = 16807\), then the value of \(7^{(a-3)}\) is:
Answer: A
\[ 16807 = 7^{5} \quad\Longrightarrow\quad a = 5 \]
\[ 7^{a-3} = 7^{2} = 49 \]
Equivalently \( \tfrac{16807}{343} = 49 \).
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\(\frac{(243)^{n/5}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}=?\)
Answer: C
Given Expression = \(\frac{(243)^{(n/5)}\times 3^{2n+1}}{9^{n}\times 3^{n-1}}\)
\(=\frac{(3^{5})^{(n/5)}\times 3^{2n+1}}{(3^{2})^{n}\times 3^{n-1}}\)
\(=\frac{(3^{5\times(n/5))}\times 3^{2n+1}}{(3^{2n}\times 3^{n-1})}\)
\(=\frac{3^{n\times3^{2n+1}}}{3^{2n}\times3^{n-1}}\)
\(=\frac{3^{(n+2n+1)}}{3^{(2n+n-1)}}\)
\(=\frac{3^{3n+1}}{3^{3n-1}}\)
\(=3^{(3n+1-3n+1)}=3^{2}=9\)
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Solve for \(m\) if \(49(7^m) = 343^{3m+6}\)
Answer: B
Exp: \(49(7^m) = 343^{3m+6} \)
\(\Rightarrow 7^2 7^m = (7^3)^{3m+6} \)
\(\Rightarrow 7^{2+m} = 7^{9m+18}\)
Equating powers of 7 on both sides,
\(\Rightarrow m + 2 = 9m + 18\)
\(\Rightarrow - 16 = 8m\)
\(\Rightarrow m = -2\)
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If \(x = \sqrt{3} + \sqrt{2}\), what is the value of \(\dfrac{1}{x}\) ?
Answer: C
Correct answer: \(\sqrt{3} - \sqrt{2}\)
Rationalise the denominator by multiplying above and below by the conjugate.
\[\frac{1}{\sqrt{3}+\sqrt{2}} \times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{\sqrt{3}-\sqrt{2}}{3-2} = \sqrt{3}-\sqrt{2}\]
The denominator becomes \(a^2 - b^2\), which clears both surds at once.
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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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The value of \([(10)^{150} \div (10)^{146}]\)
Answer: B
\((10)^{150} \div (10)^{146} \)
\(= \frac{10^{150}}{10^{146}}\)
\(= 10^{150-146} \)
\(= 10^{4} = 10000\)
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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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\((\frac{1}{216})^{-\frac{2}{3}}\div(\frac{1}{27})^{-\frac{4}{3}}=?\)
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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\(\frac{1}{1+a^{(n-m)}}+\frac{1}{1+a^{(m-n)}}=?\)
Answer: B
\(\frac{1}{1+a^{(n-m)}}+\frac{1}{1+a^{(m-n)}}=\frac{1}{(1+\frac{a^{n}}{a^{m}})}+\frac{1}{(1+\frac{a^{m}}{a^{n}})}\)
\(=\frac{a^{m}}{(a^{m}+a^{m})}+\frac{a^{m}}{(a^{m}+a^{n})}\)
\(=\frac{(a^{m}+a^{n})}{(a^{m}+a^{n})}\)
=1
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