Simplify and write the exponential form with the negative exponent:
${( - 7)^3} \times {\left( {\dfrac{1}{{ - 7}}} \right)^{ - 9}} \div {\left( { - 7} \right)^{10}}$.

Hint: We have to simplify the given expression and then we have to write the expression in the form of a negative exponent. We know that the negative sign on an exponent means the reciprocal.
 It means that if we have ${( - a)^{ - m}}$, then it can be written as
${\left( {\dfrac{1}{{ - a}}} \right)^m}$. So we will apply this and some of the basic exponential formulas to solve this.

Complete step-by-step solution:
Here we have been given
 ${( - 7)^3} \times {\left( {\dfrac{1}{{ - 7}}} \right)^{ - 9}} \div {\left( { - 7} \right)^{10}}$.
We will first simplify the negative exponential power i.e.
. ${\left( {\dfrac{1}{{ - 7}}} \right)^{ - 9}}$.
So we will try to reciprocate term which is inside the bracket i.e.
${\left( {\dfrac{1}{{\dfrac{1}{{ - 7}}}}} \right)^9}$.
It gives us value
 ${\left( {1 \times \dfrac{{ - 7}}{1}} \right)^9} = {\left( { - 7} \right)^9}$
So our new expression is
 ${( - 7)^3} \times {( - 7)^9} \div {\left( { - 7} \right)^{10}}$.
We can see that all the bases of the powers are the same, so we will apply the division and multiplication exponential formula.
The division formula states that ${a^m} \div {a^n} = {a^{m - n}}$
By comparing here we have
 $a = - 7,m = 9,n = 10$
By applying the formula we can write
 ${( - 7)^3} \times {( - 7)^{9 - 10}}$.
It gives us
 ${( - 7)^3} \times {( - 7)^{ - 1}}$.
Since we have product left, we will apply the formula product rule of exponent. It says that in multiplication if the two powers have the same base, then we can add the exponents.
We can write this as
${(a)^m} \times {(a)^n} = {(a)^{m + n}}$.
We will compare the formula with expression and we have
$a = - 7,m = 3,n = - 1$
So we can write
${( - 7)^{3 + ( - 1)}} = {( - 7)^2}$
Now we have to write the expression in negative exponent, it says that
 ${(a)^m} = {\left( {\dfrac{1}{a}} \right)^{ - m}}$
Similarly by applying this we can write
 ${( - 7)^2} = {\left( {\dfrac{1}{{ - 7}}} \right)^{ - 2}}$
Hence the required answer is ${\left( {\dfrac{1}{{ - 7}}} \right)^{ - 2}}$.

Note: We should note that a positive exponent means repeated multiplication of the base with base and a negative exponent means repeated division of the base by the base. So when we move the negative exponent to the denominator, it becomes positive.
So if we have ${x^{ - 3}}$, it can be written as $\dfrac{1}{{{x^3}}}$. We can see that the negative exponent changes to positive.