Question

evaluate
$\frac{{\left(\frac{1}{2}\right)}^5-{\left(\frac{1}{2}\right)}^6}{{\left(\frac{1}{2}\right)}^3-{\left(\frac{1}{2}\right)}^5}$

Answer 1

$\frac{{\left(\frac{1}{2}\right)}^5-{\left(\frac{1}{2}\right)}^6}{{\left(\frac{1}{2}\right)}^3-{\left(\frac{1}{2}\right)}^5}$

$={\left(\frac{1}{2}\right)}^5[1-\frac{1}{2}]$

${\left(\frac{1}{2}\right)}^3[1-{\left(\frac{1}{2}\right)}^2]$

$=\frac{{\left(\frac{1}{2}\right)}^{5-3}(1-\frac{1}{2})}{(1-\frac{1}{2})(1+\frac{1}{2})}\because a^2-b^2=(a-b)(a+b)$

$=\frac{{\left(\frac{1}{2}\right)}^2}{\left(\frac{3}{2}\right)}$

$\Rightarrow \frac{1}{4}\times \frac{2}{3}\Rightarrow \frac{1}{6}$