Express ⎛⎜⎝512+918⎞⎟⎠÷⎛⎜⎝512−918⎞⎟⎠ as an equivalent fraction with a rational denominator.
To rationalise the denominator, which is equal to 512−314, put 512=x and 314=y; then since x4−y4=(x−y)(x3+x2y+xy2+y4)
The required factor is 532+522.314+512.324+334;
and the rational denominator is 542−344=52−3=22
∴ the expression =(512+314)(532+522.314+512.324+334)22
=542+2.532.314+2.522324+2.512.324+34422
=14+532.314+5.312+512.32411