Question

Which is greater $7^{92}$ or $8^{91}$?

Answer 1

$\frac{\left(7^{92}\right)}{\left(8^{91}\right)}$

$=(\frac{7}{8}{)}^{91}\times 7$

Now, $\frac{7}{8}=0.875$

As we know that, $0.875\times 0.875\approx 0.75(0.9\times 0.9=0.81)$

$0.875\times 0.75\approx 0.6$

Next, $0.875\times 0.6\approx 0.5$

The point is every subsequent multiplication of $0.875$ reduces the numerator by digit

clearly after 4 more such multiplication, we land at 0.1

Therefore, the total numerator$=0.1\times 7=0.7$

Which is already less than 1

Hence $(8{)}^{91}>7^{92}$