How do you simplify #\frac { z y ^ { 5} } { z ^ { 7} y ^ { 3} }#?

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Answer 1 · 2017-07-31T02:05:46

Expression #= (y/z^3)^2#

Here we will use three rules of exponents as follows:

(i) #1/a = a^-1#

(ii) #a^mxxa^n = a^(m+n)#

(iii) #(a^m)^n = a^(mxxn)#

Expression #= (zy^5)/(z^7y^3)#

#= z^1/z^7 xx y^5/y^3#

Apply (i) & (ii)

#= z^(1-7) xx y^(5-3)#

#= z^-6 xx y^2#

Reverse (i)

#= y^2 xx 1/z^6 = y^2/z^6#

Apply (iii)

#= (y/z^3)^2#

Answer 2 · 2017-07-31T02:14:52

#y^2/z^6# or #y^2z^-6#

A] One way is to subtract the smaller exponent from the larger exponent of the same number.

#(zy^5)/(z^7y^3)=(y^(5-3))/(z^(7-1))=y^2/z^6=y^2z^-6#

B] Another way is to expand the exponents and cancel the common ones in the numerator and denominator.

#(z*y*y*y*y*y)/(z*z*z*z*z*z*z*y*y*y)=(cancelz*cancely*cancely*cancely*y*y)/(cancelz*z*z*z*z*z*z*cancely*cancely*cancely)=(y*y)/(z*z*z*z*z*z)=y^2/z^6#

We can write this as #y^2/z^6# or bring the denominator up by changing the exponent into a negative.

#y^2z^-6#