How do you simplify #a^ { - 8} b ^ { - 4} c ^ { 0} d \cdot a ^ { 9} b ^ { 3} c ^ { - 1} d \cdot a b ^ { 5} c ^ { - 3} d#?

Recall that for any numbers a and b...

#a^(-b) = 1/a^b, a^0 = 1, x^a/x^b = x^(a-b), x^a * x^b = x^(a+b)#

Thus our expression can be written:

#a^(−8)b^(−4)c^0d⋅a^9b^3c^(−1)d⋅ab^5c^(−3)d = d/(a^8b^4) (a^9b^3d)/ c * (ab^5d) / c^3#

#=(a^10b^8d^3)/(a^8b^4c^4) = (a^2b^4d^3)/c^4#

This is just one big long multiplication problem

1) Clarify the problem by writing it as a long string of factors

#a^-8 * b^-4 * c^0 * d^1 ⋅ a^9 * b^3 * c^-1 * d^1 ⋅ a^1 * b^5 * c^-3 * d^1#

2) Simplify the problem by flipping the factors with negative exponents into the denominator and reversing their signs to plus

#(c⁰ * d^1 ⋅ a^9 * b^3 * d^1 ⋅ a^1 * b^5 * d^1)/ (a^8 * b^4 * c^1 *c^3)#

3) Collect like terms
(Note: #n^0 = 1#, so #c ⁰# just drops out.)

#( [a^9 ⋅ a^1] ⋅ [b^3 * b^5] * [d^1 * d^1 * d^1])/ ([a^8] * [b^4] * [c^1 *c^3])#

4) Clear the parentheses by multiplying the like terms
To multiply exponents, you add them

#(a^10 * b^8 * d^3) / (a^8 * b^4 * c^4)#

5) Reduce the fraction to lowest terms by canceling.
To divide exponents, you subtract

After you have subtracted the exponents in the denominator from the exponents of their like terms in the numerator, you will have this:

#(a^2 * b^4 * d^3)/c^4# #larr# answer

Answer:
The expression simplifies to

#(a^2 * b^4 * d^3)/c^4# #larr# answer

Sometimes it is desirable for the answer to be written without the hassle of a denominator.

To get #c^4# out of the denominator, flip it into the numerator and reverse its sign to negative.

#a^2# #b^4# #c^-4# #d^3# #larr# same answer