Question

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

Answer 1

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

Explanation:

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

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

Explanation:

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`