Question

Find the difference between ${{3}^{{{3}^{3}}}}-{{\left( {{3}^{3}} \right)}^{3}}$?
(a) ${{3}^{27}}-{{3}^{9}}$
(b) 0
(c) ${{27}^{3}}-{{3}^{27}}$
(d) ${{3}^{18}}\left( {{3}^{9}}-1 \right)$

Answer 1

Hint: We start solving the problem by assigning the variable for the given difference of two terms. We then use one of the laws of exponents ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ for one of the terms in the given difference. We then use the condition that in term ${{a}^{{{m}^{n}}}}$, we first find the value of ${{m}^{n}}$ and then proceed to find the value of ${{a}^{{{m}^{n}}}}$ to get the value of another term in the difference. We then substitute these values in the difference to get the required result.

Complete step-by-step solution
According to the problem, we are given that we need to find the difference between the terms ${{3}^{{{3}^{3}}}}-{{\left( {{3}^{3}} \right)}^{3}}$.
Now, let us assume the difference is ‘d’.
So, we have $d={{3}^{{{3}^{3}}}}-{{\left( {{3}^{3}} \right)}^{3}}$ ---(1).
According to the law of exponents, we know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$. Let us use this property in equation (1).
So, we get $d={{3}^{{{3}^{3}}}}-{{3}^{3\times 3}}$.
$\Rightarrow d={{3}^{{{3}^{3}}}}-{{3}^{9}}$ ---(2).
We know that if we have the term as ${{a}^{{{m}^{n}}}}$, we first find the value of ${{m}^{n}}$ and then proceed to find the value of ${{a}^{{{m}^{n}}}}$. Let us use this in equation (2).
$\Rightarrow d={{3}^{27}}-{{3}^{9}}$.
So, we have found the value of difference between the terms ${{3}^{{{3}^{3}}}}-{{\left( {{3}^{3}} \right)}^{3}}$ as ${{3}^{27}}-{{3}^{9}}$.
The correct option for the given problem is (a).

Note: We should not confuse and take ${{\left( {{3}^{3}} \right)}^{3}}={{3}^{{{3}^{3}}}}$, as both were different which can be explained as follows. We should know that in term ${{a}^{{{m}^{n}}}}$, the term ‘m’ is raised to the power ‘n’ and the term ‘a’ is raised to the power of ${{m}^{n}}$, whereas in ${{\left( {{a}^{m}} \right)}^{n}}$ the term ${{a}^{m}}$ is raised to the power ‘n’ for which the values will be much different. Similarly, we can expect problems to find which is greater ${{3}^{{{3}^{3}}}}$ or ${{\left( {{3}^{3}} \right)}^{3}}$.