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How do you find the equivalent exponential expression ${{\left( {{7}^{3}} \right)}^{2}}$?

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: It is an expression of power of another power. So, multiply both the powers first. You can use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$. Then do the necessary simplification, if required to obtain the required solution.

Complete step by step answer:
An expression containing a power of another power, we can write it as a single power with the exponent equal to the multiplication of both the powers of the given expression.
So if we have a base ‘a’, the first power as ‘m’ and the second power over ‘m’ as ‘n’ in the form ${{\left( {{a}^{m}} \right)}^{n}}$, then we can simplify it as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}={{a}^{mn}}$
Now, considering our expression ${{\left( {{7}^{3}} \right)}^{2}}$
Here, ‘3’ is the first power and ‘2’ is another power over ‘3’. So, to write it as a single power of ‘7’, we have to multiply both the powers.
Multiplying both the powers, we get
\[\begin{align}
  & \Rightarrow {{7}^{3\times 2}} \\
 & \Rightarrow {{7}^{6}} \\
 & \Rightarrow 117649 \\
\end{align}\]
This is the required solution of the given question.

Note: The powers should be multiplied to convert the expression to a single exponent of ‘7’. Then it should be written in maximum simplified form. For example the above expression could also be written as the single exponent of ‘7’ i.e. ${{7}^{6}}$. But the numeric value of ${{7}^{6}}=117649$ is the appropriate answer to the given question.