Question

By what number should $3^{-4}$ be multiplied so that the product is 729?

• A. $3^{10}$
• B. $3^{11}$
• C. $3^{12}$
• D. $3^{13}$

Answer 1

The correct option is A $3^{10}$

Let the number be $X$.
$⟹3^{-4}\times X=729$
we can write 729 as
$\begin{array}{cc}3& 729\\ 3& 243\\ 3& 81\\ 3& 27\\ 3& 9\\ 3& 3\\ & 1\end{array}$
$729=3\times 3\times 3\times 3\times 3\times 3=3^6$
Now putting $3^6$ in place of 729
$⟹3^{-4}\times X=3^6$
$⟹X=\frac{3^6}{3^{-4}}$
By the Laws of Exponents, we know that
$\frac{a^m}{a^n}=a^{m-n}$
$⟹X=\frac{3^6}{3^{-4}}=3^{6-(-4)}=3^{6+4}=3^{10}$
So the number to be multiplied by $3^{-4}$ is $3^{10}$ so that the product is 729.