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Algebra

Find the integer closest to \(\frac { 1 }{ \sqrt [ 4 ]{ 5^{ 4 }+1 } -\sqrt [ 4 ]{ 5^{ 4 }-1 } } \)

I wanted to post this as a question but found multiple answers Can anyone help me with this question? I think the answer might be \(1\) OR \(0\) OR \(250\) which one is correct?

Note by Ankit Vijay
2 years, 6 months ago

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Consider \( \LARGE \frac {1}{ \sqrt[4]{A} - \sqrt[4]{B} } \)

Multiply numerator and denominator by \( \large \sqrt[4]{A} + \sqrt[4]{B} \), apply difference of squares: \(x^2 - y^2 = (x-y)(x+y) \), we have

\[ \LARGE \frac { \sqrt[4]{A} + \sqrt[4]{B} }{ \sqrt[2]{A} - \sqrt[2]{B} } \]

Now this time, multiply numerator and denominator by \( \large \sqrt[2]{A} + \sqrt[2]{B} \)

\[ \LARGE \frac { \left ( \sqrt[4]{A} + \sqrt[4]{B} \right ) \left ( \sqrt[2]{A} + \sqrt[2]{B} \right ) }{ A-B } \]

Now set \(A = 5^4 + 1, B = 5^4 - 1 \), the denominator equals to \(2\)

And because \( A \approx 5^4, B \approx 5^4 \), then \( \large \sqrt[4]{A} \approx 5, \sqrt[4]{B} \approx 5, \sqrt[2]{A} \approx 25, \sqrt[2]{B} \approx 25 \)

So the expression is appproximately close to \( \frac {(5+5)(25+25)}{2} = \boxed{250} \)


Alternatively, you can use binomial expansion, for \( |x| <1 \), we have \( (1 + x)^n \approx 1 + nx \)

Because \( \large \sqrt[4]{5^4 + 1} = \sqrt[4]{5^4 \left ( 1 + \frac {1}{5^4} \right ) } = 5 \cdot \sqrt[4]{ 1 + \frac {1}{5^4} } \), likewise \( \large \sqrt[4]{5^4 - 1} = 5 \cdot \sqrt[4]{ 1 - \frac {1}{5^4} } \)

So the expression equals to \( \large \left ( 5(1 + x)^n - 5(1-x)^n \right )^{-1} \)

Substitution of \( \large x = \frac {1}{5^4}, n = \frac {1}{4} \) gives \( \large \frac {1}{10nx} = \boxed{250} \) Pi Han Goh · 2 years, 6 months ago

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@Pi Han Goh Beautiful answer... Ankit Vijay · 2 years, 6 months ago

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are you nuts?? Abhinav Jangir · 1 year, 10 months ago

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