I'm going to write this in a little differently because it is how I am interpreting your question, hopefully I am reading the same thing you trying to get answered.

(a^{2}·b^{2}·c^{-3}) ^{-4}· a^{4}·c^{-3} ÷ (2a^{-2}· b^{3})

We need to know rules for exponents. Combine like exponents (the exponents for x's together, etc.)

an exponent raised to an exponent: multiply the exponents (x^{2})^{4} = x^{8}

an exponent multiplied to and exponent: add the exponents x^{2}·x^{4} = x^{6}

an exponent divided by an exponent: sub the bottom from the top x^{2}/x^{4} = x^{-2}

a positive exponent means it belongs on the top of a fraction

a negative exponent means it belongs on the bottom of a fraction

Lets break it into two pieces and put back together at end

#1: (a^{2}·b^{2}·c^{-3}) ^{-4}· a^{4}·c^{-3}

distribute the -4 to the ( )

a^{-8} b^{-8} c^{12} · a^{4}c^{-3}

add exponents (because all the letters are being multiplied together)

a^{-4} b^{-8} c^{9}

put the exponents on top and bottom of a fraction based on + or - sign or exponent

c^{9}/a^{4}b^{8}

we will come back to this in a second

#2

2a^{-2}· b^{3}

put the exponents on top and bottom of a fraction based on + or - sign or exponent

2b^{3}/a^{2}

#1 and #2 together

(c^{ 9}/a^{4}b^{ 8})÷(2b^{ 3}/a^{2})

to divide fractions we flip the bottom one (the reciprocal) and multiply the fractions together

(c ^{9}/a^{4}b ^{8})×(a^{ 2}/2b
^{3})

multiply across the top to get new numerator, and across bottom to get new denominator

a^{2}c ^{9}/2a^{4}b^{8}b^{ 3}

add, sub, multiply as needed to simplify further

a: 2-4 = -2

b: 8+3 = 11

c: 9-0 = 9

c ^{9}/2a^{2}b^{11}

## Comments

the two should be in the denominator here.

Good catch. I will correct my answer.