What are the rules for simplifying fractions with variables?

Rules for simplifying fractions with variables:

1. Factor the numerator and denominator of the fraction;

2. Find common factors to both the numerator and denominator;

3. Cancel down all the common factors.

2. Find common factors to both the numerator and denominator;

3. Cancel down all the common factors.

Problem 1: Simplify the fraction with variables

10a |
---|

14a^{2} |

Solution.

Step by step.

1. Factor the numerator and denominator of the fraction.

The prime factorization of 10

2 * 5

The prime factorization of 14

2 * 7

2. Find common factors to both the numerator and denominator.

We have

10a | = |
---|---|

14a^{2} |

2 * 5 * a |
---|

2 * 7 * a^{2} |

The common factor is 2a.

3. Cancel down all the common factors.

Cancel the common factor of 2a

2 * 5 * a | = |
---|---|

2 * 7 * a * a |

2a * 5 | = |
---|---|

2a * 7 * a |

5 |
---|

7a |

4. The answer is

10a | = | 5 |
---|---|---|

14a^{2} | 7a |

The problem is solved.

Problem 2: Simplify the fraction with variables

4 - a^{2} |
---|

2 + a |

Solution.

1. Factor the numerator of the fraction.

Factor the numerator

4 - a^{2} =

2^{2} - a^{2} =

(2 + a)(2 - a)

2

(2 + a)(2 - a)

2. Find common factors to both the numerator and denominator.

We have

4 - a^{2} | = |
---|---|

2 + a |

(2 + a)(2 - a) |
---|

2 + a |

The common factor is (2 + a).

3. Cancel down all the common factors.

Cancel the common factor of (2 + a)

(2 + a)(2 - a) | = |
---|---|

2 + a |

(2 + a) * (2 - a) | = |
---|---|

2 + a |

2 - a |

4. The answer is

4 - a^{2} | = 2 - a |
---|---|

2 + a |

The problem is solved.