Question

If ${\left(2000\right)}^{10}=1.024\times {10}^k$, then the value of $k$ is

• A. $33$
• B. $30$
• C. $34$
• D. $31$

Answer 1

The correct option is A

$33$

Explanation for the correct option:

Finding the value of $k$:

$\begin{array}{l}{\left(2000\right)}^{10}=1.024\times {10}^k\\ \Rightarrow {\left(16\times 125\right)}^{10}=1.024\times {10}^k\\ \Rightarrow {\left(2^4\times 5^3\right)}^{10}=1024\times {10}^{k-3}\\ \Rightarrow \left(2^{40}\times 5^{30}\right)=2^{10}\times {10}^{k-3}\left(\because a^m\times b^m={\left(ab\right)}^m\right)\\ \Rightarrow 2^{10}\left(2^{30}\times 5^{30}\right)=2^{10}\times {10}^{k-3}\\ \Rightarrow \left(2^{30}\times 5^{30}\right)={10}^{k-3}\\ \Rightarrow \left({10}^{30}\right)={10}^{k-3}\left(\because a^m\times b^m={\left(ab\right)}^m\right)\\ \Rightarrow 30=k-3\left(\text{as the base is same on both sides}\right)\\ \Rightarrow k=33\end{array}$

Hence, the correct option is (A)