Dividing Monomials

Example 1. Simplify $$${\left({14}{{x}}^{{5}}\right)}\div{\left({7}{{x}}^{{2}}\right)}$$$.

Since fraction denotes division, then it is better (for visual perception) to rewrite $$${\left({14}{{x}}^{{5}}\right)}\div{\left({7}{{x}}^{{2}}\right)}$$$ as $$$\frac{{{14}{{x}}^{{5}}}}{{{7}{{x}}^{{2}}}}$$$.

$$$\frac{{{14}{{x}}^{{5}}}}{{{7}{{x}}^{{2}}}}=$$$

$$$=\frac{{14}}{{7}}\cdot\frac{{{{x}}^{{5}}}}{{{{x}}^{{2}}}}=$$$ (multiplication of fractions in reverse direction: $$$\frac{{{a}\cdot{c}}}{{{b}\cdot{d}}}=\frac{{a}}{{b}}\cdot\frac{{c}}{{d}}$$$)

$$$={2}\frac{{{{x}}^{{5}}}}{{{{x}}^{{2}}}}=$$$ (simplify coefficient)

$$$={2}{{x}}^{{{5}-{2}}}=$$$ (rule for subtracting exponents)

$$$={2}{{x}}^{{3}}$$$ (simplify).

Answer: $$${\left({14}{{x}}^{{5}}\right)}{\left({7}{{x}}^{{2}}\right)}={2}{{x}}^{{3}}$$$.

Example 2. Simplify $$$\frac{{\frac{{1}}{{2}}{{x}}^{{4}}{{y}}^{{5}}{z}}}{{-{7}{{z}}^{{3}}{x}{{y}}^{{3}}}}$$$.

$$$\frac{{\frac{{1}}{{2}}{{x}}^{{4}}{{y}}^{{5}}{z}}}{{-{7}{{z}}^{{3}}{x}{{y}}^{{3}}}}=$$$

$$$=\frac{{\frac{{1}}{{2}}}}{{-{7}}}\cdot\frac{{{{x}}^{{4}}}}{{{x}}}\cdot\frac{{{{y}}^{{5}}}}{{{{y}}^{{3}}}}\cdot{\left(\frac{{z}}{{{z}}^{{3}}}\right)}$$$ (break down fraction)

$$$=-\frac{{1}}{{14}}\cdot{{x}}^{{{4}-{1}}}\cdot{{y}}^{{{5}-{2}}}\cdot{{z}}^{{{1}-{3}}}=$$$ (rule for subtracting exponents).

$$$=-\frac{{1}}{{14}}{{x}}^{{3}}{{y}}^{{3}}{{z}}^{{-{2}}}$$$ (simplify).

You can leave answer as it is, but, in most cases, teachers don't allow negative exponents, so you need to get rid of negative exponents.

$$$=-\frac{{1}}{{14}}{{x}}^{{3}}{{y}}^{{3}}\frac{{1}}{{{{z}}^{{2}}}}=$$$ (get rid of negative exponent)

$$$=-\frac{{{{x}}^{{3}}{{y}}^{{3}}}}{{{14}{{z}}^{{2}}}}$$$ (write more compactly, using rule for multiplying fractions)

Answer: $$${\left(\frac{{1}}{{2}}{{x}}^{{4}}{{y}}^{{5}}{z}\right)}\div{\left(-{7}{{z}}^{{3}}{x}{{y}}^{{3}}\right)}=-\frac{{{{x}}^{{3}}{{y}}^{{3}}}}{{{14}{{z}}^{{2}}}}$$$.

Example 3. Divide $$${\left({4}{{x}}^{{3}}{{y}}^{{2}}\right)}\div{\left({3}{{z}}^{{5}}{y}{{x}}^{{3}}\right)}\div{\left(-{2}{{y}}^{{2}}{z}{x}\right)}$$$.

You need to do it step by step.

First divide $$${\left({4}{{x}}^{{3}}{{y}}^{{2}}\right)}\div{\left({3}{{z}}^{{5}}{y}{{x}}^{{3}}\right)}$$$: $$$\frac{{{4}{{x}}^{{3}}{{y}}^{{2}}}}{{{3}{{z}}^{{5}}{y}{{x}}^{{3}}}}=\frac{{{4}{y}}}{{{3}{{z}}^{{5}}}}$$$.

Now, divide resulting fraction by third monomial (using rule for dividing fractions): $$$\frac{{{4}{y}}}{{{3}{{z}}^{{5}}}}\div{\left(-{2}{{y}}^{{2}}{z}{x}\right)}=\frac{{\frac{{{4}{y}}}{{{3}{{z}}^{{5}}}}}}{{-{2}{{y}}^{{2}}{z}{x}}}=\frac{{{4}{y}}}{{{3}{{z}}^{{5}}\cdot{\left(-{2}{{y}}^{{2}}{z}{x}\right)}}}=\frac{{2}}{{{3}{x}{y}{{z}}^{{6}}}}$$$.

Answer: $$${\left({4}{{x}}^{{3}}{{y}}^{{2}}\right)}\div{\left({3}{{z}}^{{5}}{y}{{x}}^{{3}}\right)}\div{\left(-{2}{{y}}^{{2}}{z}{x}\right)}=\frac{{2}}{{{3}{x}{y}{{z}}^{{6}}}}$$$.