Question

If ( 13/4) whole to the power 4 - (13/3) whole to the power 4 ÷ ( 13/4)² - (13/3 )² = ( 13a÷12 )², find a.

Answer 1

$\frac{ (\frac{13}{4})^4 - (\frac{13}{3})^4 }{ (\frac{13}{4})^2 - (\frac{13}{3})^2 } = (\frac{13a}{12})^2\\ \\ \frac{ [(\frac{13}{4})^2 + (\frac{13}{3})^2][(\frac{13}{4})^2 - (\frac{13}{3})^2] }{ (\frac{13}{4})^2 - (\frac{13}{3})^2 } = (\frac{13a}{12})^2\\ \\ (\frac{13}{4})^2 + (\frac{13}{3})^2=(\frac{13a}{12})^2$

$[ \frac{13}{4} + \frac{13}{3} ]^2 - 2 \times \frac{13}{4} \times \frac{13}{3} = (\frac{13a}{12})^2\\ \\ (\frac{13 \times 3 + 13 \times 4}{12})^2 - \frac{2 \times 169}{12} = \frac{169a^2}{144} \\ \\\frac{169 \times 7^2}{144} - \frac{24 \times 169}{144} = \frac{169a^2}{144}\\ \\169(7^2-24)=a^2\\ \\a^2 = 169 \times 25\\ \\a = \sqrt{169 \times 25} = \boxed{65}$