We start with the expression
[(−25a2b2)3⋅(−25a2b2)7]3.
Using the product of powers rule, we combine the exponents:
(−25a2b2)3+7=(−25a2b2)10.
Now applying the power of a power rule:
(−25a2b2)10⋅3=(−25a2b2)30.
Step 2: Simplifying the Denominator
Next, we simplify the denominator:
[(−25a2b2)4⋅(−25a2b2)3]4.
Again, using the product of powers rule:
(−25a2b2)4+3=(−25a2b2)7.
Now applying the power of a power rule:
(−25a2b2)7⋅4=(−25a2b2)28.
Step 3: Dividing the Expressions
Now we divide the simplified numerator by the simplified denominator:
(−25a2b2)28(−25a2b2)30.
Using the quotient of powers rule:
(−25a2b2)30−28=(−25a2b2)2.
Step 4: Calculating the Final Result
Calculating the final expression:
(−25)2⋅(a2)2⋅(b2)2=425a4b4.
Thus, the numerical value of the expression is 100 when evaluated, leading to the final result.