Answer
$5.98\times10^{24}kg$
Work Step by Step
We can calculate the speed of the Moon by $v=\frac{2\pi r}{T}$ and $v=\sqrt{\frac{GM_{earth}}{r}}$. Equate both expressions and solve for mass of Earth.
$$v=\frac{2\pi r}{T}=\sqrt{\frac{GM_{Earth}}{r}}$$
$$M_{Earth}=\frac{4\pi^2r^3}{GT^2}=\frac{4\pi^2(3.84\times10^8m)^3}{(6.67\times10^{-11})[(27.4days)(86400\frac{seconds}{day})]^2}$$
$$=5.98\times10^{24}kg$$
