Question

How do you simplify ${{6}^{-5}}$?

Answer 1

Hint: In this problem we need to simplify the given value which is ${{6}^{-5}}$. For this we will use the well-known exponential formula which is ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$. So, we will use this exponential formula and write the value of the ${{6}^{-5}}$. Now in the denominator we have ${{6}^{5}}$. To calculate the value of ${{6}^{5}}$ we will multiply the $6$ with itself $5$ times and substitute this calculated value in the obtained fraction to get the required result.

Complete step by step solution:
Given that, ${{6}^{-5}}$.
We can observe that the above given value is in the form of ${{a}^{-m}}$. In exponentials we have the formula ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$. Applying this formula in the above equation, then we will get
$\Rightarrow {{6}^{-5}}=\dfrac{1}{{{6}^{5}}}...\left( \text{i} \right)$
In the above equation we have the denominator as ${{6}^{5}}$. To calculate the value of ${{6}^{5}}$ we are going to multiply the $6$ with itself $5$ times. First multiplying $6$ with $6$, then we will get
$\Rightarrow 6\times 6=36$
Multiplying the above value with again $6$, then we will have
$\begin{align}
  & \Rightarrow 6\times 6\times 6=36\times 6 \\
 & \Rightarrow 6\times 6\times 6=216 \\
\end{align}$
Again, multiplying the $6$ to the above values, then we will get
$\begin{align}
  & \Rightarrow 6\times 6\times 6\times 6=216\times 6 \\
 & \Rightarrow 6\times 6\times 6\times 6=1296 \\
\end{align}$
Again, multiplying the above value with the $6$, then we will have
$\begin{align}
  & \Rightarrow 6\times 6\times 6\times 6\times 6=1296\times 6 \\
 & \Rightarrow 6\times 6\times 6\times 6\times 6=7776 \\
\end{align}$
From the above equation we can say that we have multiplied $6$ with itself $5$ time, hence the value of ${{6}^{5}}$ will be ${{6}^{5}}=7776$. Substituting this value in the equation $\left( \text{i} \right)$, then we will get
$\Rightarrow {{6}^{-5}}=\dfrac{1}{7776}$

Note: We can also stop the solution after getting the result as ${{6}^{-5}}=\dfrac{1}{{{6}^{5}}}$. There is no need to calculate the value of ${{6}^{5}}$ and substitute it in the above calculated value. Another possible way to write the final answer is to compute it in decimals and round it off as 0.00013. So, if this was an MCQ question, we could have chosen the appropriate answer, which could have been in any of these forms.