Question

How do you evaluate ${{6}^{3}}$?

Answer 1

Hint: In this problem we need to calculate the value of ${{6}^{3}}$. We can observe that the given number is in exponential form with base $6$ and power $3$. Now we will first factorise the base $6$ and we will write base as the product of the it’s factors. Now we will apply the exponential rule ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$. After applying this formula, we will calculate the values that are required to simplify the above equation. After simplifying the obtained equation, we will get our required result.

Complete step-by-step solution:
Given that, ${{6}^{3}}$.
The above given value is in exponential form with base $6$ and power $3$.
Considering the base of the given value which is $6$. Factorising the base value $6$.
When we divide the value $6$ with $2$ then we will get $3$ as remainder. So, we can write the digit $6$ as $6=2\times 3$.
Powering the above equation with $3$ on both sides, then we will get
$\Rightarrow {{6}^{3}}={{\left( 2\times 3 \right)}^{3}}$
Using the exponential formula ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$ in the above equation, then we will get
$\Rightarrow {{6}^{3}}={{2}^{3}}\times {{3}^{3}}$
From the above equation we can say that to find the required value we need to calculate the values of ${{2}^{3}}$ and ${{3}^{3}}$.
We know that the values ${{2}^{3}}=8$, ${{3}^{3}}=27$. Substituting these values in the above equation, then we will get
$\begin{align}
  & \Rightarrow {{6}^{3}}=8\times 27 \\
 & \Rightarrow {{6}^{3}}=216 \\
\end{align}$
Hence the value of the ${{6}^{3}}$ is $216$.

Note: We can also directly find the value of the given number. We can use the exponential formula ${{a}^{n}}=a\times a\times a\times a\times a\times ......\text{ n times}$. From this formula we can write ${{6}^{3}}$ as
$\Rightarrow {{6}^{3}}=6\times 6\times 6$
Multiplying the above values, we will get
$\Rightarrow {{6}^{3}}=216$
From both the methods we got the same result.