Express the following numbers using the exponential form:
1. $-\dfrac{32}{343}$
2. $\dfrac{2187}{3125}$
3. 512

Hint: We here need to express the given fractions and numbers in exponential form. For this, we need to know that whenever we express any number in exponential form, we express it in the powers of prime numbers. So here also, we will do prime factorization of the given numbers. Then we will write the numbers in the form of its factors and then we will write it in its exponential form.

Complete step-by-step solution
Here we have been asked to express some numbers in exponential form. We know that when we need to express numbers in exponential form, we express them in the form of powers of prime numbers. Thus, we will have to factorize all the numbers. This is done as:
1. $-\dfrac{32}{343}$
We will first factorize 32.
It is done as follows:
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Hence, we can write 32 as:
\[\begin{align}
  & 32=2\times 2\times 2\times 2\times 2 \\
 & \Rightarrow 32={{2}^{5}} \\
\end{align}\]
Now, we will factorise 343.
It is done as follows:
$\begin{align}
  & 7\left| \!{\underline {\,
  343 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 343 as:
$\begin{align}
  & 343=7\times 7\times 7 \\
 & \Rightarrow 343={{7}^{3}} \\
\end{align}$
Hence, $-\dfrac{32}{343}$ is written as:
$-\dfrac{32}{343}=-\dfrac{{{2}^{5}}}{{{7}^{3}}}$
Now, we know that $\dfrac{1}{x}={{x}^{-1}}$, thus we can say that:
$\begin{align}
  & -\dfrac{32}{343}=-\dfrac{{{2}^{5}}}{{{7}^{3}}} \\
 & \Rightarrow -\dfrac{32}{343}=-{{2}^{5}}{{\left( {{7}^{3}} \right)}^{-1}} \\
 & \therefore -\dfrac{32}{343}=-{{2}^{5}}{{7}^{-3}} \\
\end{align}$
 Thus, the exponential form of $-\dfrac{32}{343}$ is $-{{2}^{5}}{{7}^{-3}}$.
2. $\dfrac{2187}{3125}$
We will first factorise 2187:
$\begin{align}
  & 3\left| \!{\underline {\,
  2187 \,}} \right. \\
 & 3\left| \!{\underline {\,
  729 \,}} \right. \\
 & 3\left| \!{\underline {\,
  243 \,}} \right. \\
 & 3\left| \!{\underline {\,
  81 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 2187 as:
$\begin{align}
  & 2187=3\times 3\times 3\times 3\times 3\times 3\times 3 \\
 & \Rightarrow 2187={{3}^{7}} \\
\end{align}$
Now we will factorise 3125.
It is done as follows:
$\begin{align}
  & 5\left| \!{\underline {\,
  3125 \,}} \right. \\
 & 5\left| \!{\underline {\,
  625 \,}} \right. \\
 & 5\left| \!{\underline {\,
  125 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 3125 as:
$\begin{align}
  & 3125=5\times 5\times 5\times 5\times 5 \\
 & \Rightarrow 3125={{5}^{5}} \\
\end{align}$
Hence, we can write $\dfrac{2187}{3125}$ as:
$\begin{align}
  & \dfrac{2187}{3125}=\dfrac{{{3}^{7}}}{{{5}^{5}}} \\
 & \Rightarrow \dfrac{2187}{3125}={{3}^{7}}{{\left( {{5}^{5}} \right)}^{-1}} \\
 & \therefore \dfrac{2187}{3125}={{3}^{7}}{{5}^{-5}} \\
\end{align}$
Thus, the exponential form of $\dfrac{2187}{3125}$ is ${{3}^{7}}{{5}^{-5}}$.
3. 512
We will now factorize 512.
It is done as follows:
$\begin{align}
  & 2\left| \!{\underline {\,
  512 \,}} \right. \\
 & 2\left| \!{\underline {\,
  256 \,}} \right. \\
 & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
Thus, we can write 512 as:
$\begin{align}
  & 512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \\
 & \therefore 512={{2}^{9}} \\
\end{align}$
Thus the exponential form of 512 is ${{2}^{9}}$.

Note: For those who don’t know, prime factorization is the method of calculating the prime factors of any given number. It is shown in this question. Also, we should be very careful in this process while counting these factors as any mistake that will result in a wrong answer.