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1.)a^2/3 b^-3/4 divided by a^5/6 b^2 2.) (-2x^4y^-5) ^-3 divided by (4x^-3 y^4)^-2

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If terms are in the numerator, powers are added. If they are in denominator, powers are subtracted.

So we have a2/3-5/6 b-3/4-2 = a-1/6b-11/4 =1/(a1/6b11/4)

(-2x4y-5)-3 / (4x-3y4)-2 = (4x-3y4)2/(-2x4y-5)3 = (16x-6y8) / (-8x12y-15)

= -2x-6-12y8+15 =-2x-18y23 = -2 y23 / x18

Think of a negative exponent as really putting the number raised to the exponent on the wrong side of a fraction.  I mean by that if I have 3^-2 that is really 1/3^2.   or if I have 1/(3^-2)  that is really 3^2. to get rid of negative exponents just flip the number being raised to the negative power. But becafeful to flip exactly what is being raised to the negative exponent.

In you problem 1.  b^-3/4 is on top so move it to the bottom and drop the negative

       [a^2/3 * b^-3/4]  /  [a^5/6 * b^2]  = a^2/3   /   [a^5/6 * b^2 * b^3/4]  there is a lot more simplifying that can be done but that is getting rid of negative exponents


In your problem 2. [-2x^4 * 4y^-5]^-3   / [4x^-3 * y^4]^-2      lots going on here. DON'T try to do this all at once. Take an extra moment and an extra step or two. First let's get rid of the outside negative exponents by flipping both expression in [ ] . so we now have

[4x^-3 * y^4]^2 / [-2x^4 * 4y^-5]^3       no the outside negative exponents are gone but we need to expand-out so we can deal with the junk inside the [ ]. doing that we have (remember when you expand something inside [ ] raised to a power each term inside the [ ] are raised to that power)

4^2 * x^-6 * y^8  /  [ -2^3 * x^12 * 4^3 * y^-15 ]   = [ 16 x^-6 * y^8 ]  / [-8 x^12 * 64 y^-15]

now we can just flip the negative exponents to get rid of them that is flip  x^-6 and y^-15 so we get

[16 y^8 * y^15] / [-8 x^12 * 64 x^6]   combine like terms and cancel where you can (the numbers can be divided out by 8)

y^23 / [-32 x^18]