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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The value of \(5^{\frac{1}{4}}\times(125)^{0.25}\) is:
Answer: C
Write both factors as powers of 5.
\[ 125^{0.25} = (5^{3})^{1/4} = 5^{3/4} \]
\[ 5^{1/4} \times 5^{3/4} = 5^{1} = 5 \]
Exponents add when the bases match.
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