Find m so that ${\left( { - 3} \right)^{m + 1}} \times {\left( { - 3} \right)^5} = {\left( { - 3} \right)^7}$
(a)$2$
(b)$3$
(c)$1$
(d) none of these.

Hint: Power of a number simply provides the information regarding the number of times repetition of that number in the form of multiplication.. The rule of addition can be applied when the two exponent numbers with the same base are multiplied to each other.

Complete step-by-step answer:
The given expression is ${\left( { - 3} \right)^{m + 1}} \times {\left( { - 3} \right)^5} = {\left( { - 3} \right)^7}$
 Here, the base is the same of all the exponent numbers so we can sum up the power.
\[\begin{array}{l}
{\left( { - 3} \right)^{m + 1 + 5}} = {\left( { - 3} \right)^7}\\
{\left( { - 3} \right)^{m + 6}} = {\left( { - 3} \right)^7}
\end{array}\]
Now, the base is the same as both the numbers so the power of the numbers will be the same to satisfy the rule of equality.
Now we will equate the powers.
$\begin{array}{l}
m + 6 = 7\\
m = 7 - 6\\
m = 1
\end{array}$
Thus, the value of $m$ is $1$ and the correct answer is ${\rm{option}}\,{\rm{(c)}}$ .
So, the correct answer is “Option C”.

Note: In such types always look for arithmetic operations for the exponent numbers of the same base. If the two or more exponent numbers with the same base are in the form of division, then we can subtract powers and solve them further. When two exponent numbers are equal to each other and the base of the numbers is the same then the power of the numbers will be equal.