## Dividing Monomials

Contents

We have now seen all the properties of exponents. We’ll use them to divide monomials. Later, you’ll use them to divide polynomials.

## Example

Find the quotient: \(56{x}^{5}÷7{x}^{2}.\)

### Solution

\(56{x}^{5}÷7{x}^{2}\) | |

Rewrite as a fraction. | \(\frac{56{x}^{5}}{7{x}^{2}}\) |

Use fraction multiplication to separate the number part from the variable part. | \(\frac{56}{7}\cdot \frac{{x}^{5}}{{x}^{2}}\) |

Use the Quotient Property. | \(8{x}^{3}\) |

When we divide monomials with more than one variable, we write one fraction for each variable.

## Example

Find the quotient: \(\frac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}.\)

### Solution

\(\frac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}\) | |

Use fraction multiplication. | \(\frac{42}{-7}\cdot \frac{{x}^{2}}{x}\cdot \frac{{y}^{3}}{{y}^{5}}\) |

Simplify and use the Quotient Property. | \(-6\cdot x\cdot \frac{1}{{y}^{2}}\) |

Multiply. | \(-\frac{6x}{{y}^{2}}\) |

## Example

Find the quotient: \(\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}.\)

### Solution

\(\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}\) | |

Use fraction multiplication. | \(\frac{24}{48}\cdot \frac{{a}^{5}}{a}\cdot \frac{{b}^{3}}{{b}^{4}}\) |

Simplify and use the Quotient Property. | \(\frac{1}{2}\cdot {a}^{4}\cdot \frac{1}{b}\) |

Multiply. | \(\frac{{a}^{4}}{2b}\) |

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

## Example

Find the quotient: \(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.\)

### Solution

\(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\) | |

Simplify and use the Quotient Property. | \(\frac{2{y}^{6}}{3{x}^{4}}\) |

Be very careful to simplify \(\frac{14}{21}\) by dividing out a common factor, and to simplify the variables by subtracting their exponents.

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction.

## Example

Find the quotient: \(\frac{\left(3{x}^{3}{y}^{2}\right)\left(10{x}^{2}{y}^{3}\right)}{6{x}^{4}{y}^{5}}.\)

### Solution

Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.

\(\frac{\left(3{x}^{3}{y}^{2}\right)\left(10{x}^{2}{y}^{3}\right)}{6{x}^{4}{y}^{5}}\) | |

Simplify the numerator. | \(\frac{30{x}^{5}{y}^{5}}{6{x}^{4}{y}^{5}}\) |

Simplify, using the Quotient Rule. | \(5x\) |