Question

Among the numbers $2^{250},3^{200},4^{150}$ and $5^{100}$, the greatest number is

Answer 1

Given numbers $2^{250},3^{200},4^{150}$ and $5^{100}$
Since, the exponents values are multiple of $50$.
So, we write each given number as powers of $50$.
Thus, $2^{250}=(2{)}^{5\times 50}=(2^5{)}^{50}=(32{)}^{50}$
[Since, $\left(a^{m\times n}\right)=(a^m{)}^n$]
$3^{200}=(3{)}^{4\times 50}=(3^4{)}^{50}=(81{)}^{50}$
$5^{100}=(5{)}^{2\times 50}=(5^2{)}^{50}=(25{)}^{50}$
$4^{150}=(4{)}^{3\times 50}=(4^3{)}^{50}=(64{)}^{50}$
As the power values are same in all exponents values, now can compare their base values as follow,
$81>64>32>25$
$\Rightarrow (81{)}^{50}>(64{)}^{50}>(32{)}^{50}>(25{)}^{50}$
Therefore, $(81{)}^{50}=3^{200}$ is the greatest value among the given numbers.