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

Jul 31, 2017

Expression $= {\left(\frac{y}{z} ^ 3\right)}^{2}$

#### Explanation:

Here we will use three rules of exponents as follows:

(i) $\frac{1}{a} = {a}^{-} 1$

(ii) ${a}^{m} \times {a}^{n} = {a}^{m + n}$

(iii) ${\left({a}^{m}\right)}^{n} = {a}^{m \times n}$

Expression $= \frac{z {y}^{5}}{{z}^{7} {y}^{3}}$

$= {z}^{1} / {z}^{7} \times {y}^{5} / {y}^{3}$

Apply (i) & (ii)

$= {z}^{1 - 7} \times {y}^{5 - 3}$

$= {z}^{-} 6 \times {y}^{2}$

Reverse (i)

$= {y}^{2} \times \frac{1}{z} ^ 6 = {y}^{2} / {z}^{6}$

Apply (iii)

$= {\left(\frac{y}{z} ^ 3\right)}^{2}$

Jul 31, 2017

${y}^{2} / {z}^{6}$ or ${y}^{2} {z}^{-} 6$

#### Explanation:

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

$\frac{z {y}^{5}}{{z}^{7} {y}^{3}} = \frac{{y}^{5 - 3}}{{z}^{7 - 1}} = {y}^{2} / {z}^{6} = {y}^{2} {z}^{-} 6$

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

$\frac{z \cdot y \cdot y \cdot y \cdot y \cdot y}{z \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z \cdot y \cdot y \cdot y} = \frac{\cancel{z} \cdot \cancel{y} \cdot \cancel{y} \cdot \cancel{y} \cdot y \cdot y}{\cancel{z} \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z \cdot \cancel{y} \cdot \cancel{y} \cdot \cancel{y}} = \frac{y \cdot y}{z \cdot z \cdot z \cdot z \cdot z \cdot z} = {y}^{2} / {z}^{6}$

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

${y}^{2} {z}^{-} 6$