Question

The exponential form of $625$ is ${{5}^{k}}$ then value of $k$ is

Answer 1

Hint: In this problem we need to calculate the value of $k$ where the value of ${{5}^{k}}$ is equal to $625$. We will first write the given number $625$ in exponential form. For this we need to find the prime factors of the given number $625$. For this we will check whether the given number $625$ is divisible by the prime numbers like $2$, $3$, $5$, $7$, and so on. After finding the prime factors of the number $625$ we will write the exponential form of the number by using some exponential rules like $a\times a\times a\times a\times .....\text{ n times}={{a}^{n}}$. Now we will compare the calculated exponential form of the number $625$ with the given exponential form which is ${{5}^{k}}$. To find the value of $k$ equate the both the exponential form of the number $625$.

Complete step by step solution:
Given number $625$.
To the number $625$ in exponential form we need to calculate the prime factors of the given number. For this we are going to check whether the given number is divisible by the prime number $2$ or not. We can observe that the given number $625$ is not divisible by $2$. So now checking with the next prime number which is $3$. We can observe that the given number $625$ is not divisible by $3$ also. Now checking with the next prime number which is $5$. We can observe that the given number
$625$ is divisible by $5$ and gives $125$ as quotient. So, we can write
$625=5\times 125$
Now considering the number $125$. The number $125$ is also not divisible by $2$ and $3$. But it gives $25$ as quotient when it is divided with $5$. From we can write
$125=5\times 25$
Considering the number $25$. The number $25$ is also not divisible by $2$ and $3$. But it gives $5$ as quotient when it is divided with $5$. From we can write
$25=5\times 5$
From the all the above equation we can write the given number $625$ as
$\begin{align}
  & 625=5\times 125 \\
 & \Rightarrow 625=5\times 5\times 25 \\
 & \Rightarrow 625=5\times 5\times 5\times 5 \\
\end{align}$
Applying the exponential rule $a\times a\times a\times a\times .....\text{ n times}={{a}^{n}}$ in the above equation, then we will get exponential form of the given number $625$ as
$625={{5}^{4}}$
But in the problem, we have the exponential form of $625$ as ${{5}^{k}}$. So, comparing the calculated exponential form with the given exponential form then we will get
${{5}^{k}}={{5}^{4}}$
Equating on both sides, then we will get
$k=4$

Note: We can directly divide the given number with $5$ and calculate the exponential form in terms of $5$ only since they have given that the exponential form has only the number $5$. So, no need to check whether the given number is divisible by other prime numbers.