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({x^-1 y^4 z^-4}{y^-2 z^3 x^-5})^-1 =z^r/(x^s y^t) r the exponent of z s the exponent of x t the exponent of y

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First, let's recall some properties of exponents.

•Product of like bases:   when multiplying powers with the same base,

                                   keep the common base and add the exponents.

                                   e.g.:     am · an = am+n 

•Quotient of like bases:   when dividing powers with the same base,

                                    keep the common base and subtract the exponents.

                                    e.g.:     am/an = am-n 

•Negative exponents:   negative exponents indicate reciprocation,

                                  with the exponent of the reciprocal becoming positive.

                                  e.g.:     a-m = 1/m ;   1/a-m = am  ;   (a/b)-m = a-m/b-m = bm/am 

The problem in question is:

                [(x-1y4z-4)(y-2z3x-5)]-1 = zr/(xsyt)

Let's first work out what's inside the brackets on the left-hand side of the equation:

     (x-1y4z-4)(y-2z3x-5) = (x-1x-5)(y4y-2)(z-4z3)

                                 = (x-1+(-5))(y4+(-2))(z-4+3

                                 = (x-6)(y2)(z-1)

                                 =(x-6y2z-1)

Replace what is inside the brackets in the original equation by the above simplified term this simplified term:

        [(x-1y4z-4)(y-2z3x-5)]-1 = zr/(xsyt)

        [(x-6y2z-1)]-1 = zr/(xsyt)

Taking the reciprocal of the left-hand side of the equation, its negative exponent becomes positive:

        [(x-6y2z-1)]-1 = [(x-6y2z-1)-1]/1  

                           = 1/(x-6y2z-1)1

                           = 1/(x-6y2z-1)

                           = (1/x-6)(1/y2)(1/z-1)

Take the reciprocal of the terms with the negative exponents:

                           = (x6)(1/y2)(z1)

                           = (x6z)/y2 

Replacing the left hand side of the original equation with the term above, we get:

          (x6z)/y2 = zr/(xsyt

Now we cross-multiply to arrive at the following:

         (x6z)(xsyt) = y2zr 

Multiply terms with like bases:

         (x6xs)(yt)(z) = y2zr

         (x6+s)(yt)(z) = y2zr 

Dividing both sides of this equation by  y2zr, we get:

         [(x6+s)(yt)(z)]/(y2zr) = (y2zr)/(y2zr)

Divide the terms with like bases using the quotient of like bases property to get:

        (x6+s)(yt/y2)(z/zr) = 1

        (x6+s)(yt-2)(z1-r) = 1