There are 10 ^ 10 radioactive nuclei in a given radioactive...

Recall the relationship between the present amount of element and the initial amount of element in terms of the decay constant, $N = {N}_{0}{e}^{- \lambdat}$, now the relation between decay constant and half-life, $t\dfrac{1}{2} = \dfrac{0 . 693}{\lambda}$, $⇒ N = {N}_{0}{e}^{- \dfrac{0 . 693t}{{t}_{\dfrac{1}{2}}}}$

$⇒ N = {10}^{10}{e}^{- 0 . 35} = 7 \times {10}^{9}$.