Question

Calculate a and b if
$\sqrt{ \frac{ {7}^{2014} - {7}^{2012} }{12} } = a( {7}^{b} )$
and a is not a multiple of 7. ​

Answer 1

Notice that

$7^{2014} - 7^{2012} = 7^{2012} \bigg(7^2 - 1\bigg) = 7^{2012} \times 48 = 2^4 \times 3 \times 7^{2012}$

$12=2^2\times3$, so

$\dfrac{7^{2014}-7^{2012}}{12} = 2^2 \times 7^{2012}$

Then taking the positive square root gives

$\sqrt{\dfrac{7^{2014}-7^{2012}}{12}} = \sqrt{2^2 \times 7^{2012}} = 2\times7^{1006}$

so $\boxed{a=2}$ and $\boxed{b=1006}$.