Question

By what number should $2^{-5}$ be multiplied so that the product is 1024?

• A. $2^{16}$
• B. $2^{15}$
• C. $2^{17}$
• D. $2^{13}$

Answer 1

The correct option is B $2^{15}$

Let the number be $X$.
$⟹2^{-5}\times X=1024$
we can write 1024 as
$\begin{array}{cc}2& 1024\\ 2& 512\\ 2& 256\\ 2& 128\\ 2& 64\\ 2& 32\\ 2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$
$1024=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=2^{10}$
Now putting $2^{10}$ in place of 1024
$⟹2^{-5}\times X=2^{10}$
$⟹X=\frac{2^{10}}{2^{-5}}$
Using Laws of Exponents, we know that
$\frac{a^m}{a^n}=a^{m-n}$
$⟹X=\frac{2^{10}}{2^{-5}}=2^{10-(-5)}=2^{10+5}=2^{15}$
The number to be multiplied to $2^{-5}$ is $2^{15}$ to get the product 1024.