({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

Question

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

Answer 1

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.:     a^m · aⁿ = a^m+n 

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

                                    keep the common base and subtract the exponents.

                                    e.g.:     a^m/aⁿ = a^m-n 

•Negative exponents:   negative exponents indicate reciprocation,

                                  with the exponent of the reciprocal becoming positive.

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

The problem in question is:

                [(x^-1y⁴z^-4)(y^-2z³x^-5)]^-1 = z^r/(x^sy^t)

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))(y^4+(-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 = z^r/(x^sy^t)

        [(x^-6y²z^-1)]^-1 = z^r/(x^sy^t)

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 = z^r/(x^sy^t) 

Now we cross-multiply to arrive at the following:

         (x⁶z)(x^sy^t) = y²z^r 

Multiply terms with like bases:

         (x⁶x^s)(y^t)(z) = y²z^r

         (x^6+s)(y^t)(z) = y²z^r 

Dividing both sides of this equation by  y²z^r, we get:

         [(x^6+s)(y^t)(z)]/(y²z^r) = (y²z^r)/(y²z^r)

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

        (x^6+s)(y^t/y²)(z/z^r) = 1

        (x^6+s)(y^t-2)(z^1-r) = 1