What are the rules for simplifying algebraic fractions?

Rules for simplifying algebraic fractions:

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: Simplify the algebraic fractions

32a^{9} + 36a^{7} |
---|

36a^{7} + 32a^{7} |

Solution.

Step by step.

1. Factor the numerator and denominator of the fraction.

The prime factorization of 32

2 * 2 * 2 * 2 * 2 =

2^{5}

2

The prime factorization of 36

2 * 2 * 3 * 3 =

2^{2} * 3^{2}

2

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

We have

32a^{9} + 36a^{7} | = |
---|---|

36a^{7} + 32a^{7} |

2^{5} * a^{9} + 2^{2} * 3^{2} * a^{7} | = |
---|---|

2^{2} * 3^{2} * a^{7} + 2^{5} * a^{7} |

(2^{2} * a^{7})(2^{3} * a^{2} + 3^{2}) |
---|

(2^{2} * a^{7})(3^{2} + 2^{3}) |

The common factors are 2^{2} and a^{7}.

3. Cancel down all the common factors.

Cancel the common factors

(2^{2}a^{7})(2^{3}a^{2} + 3^{2}) | = |
---|---|

(2^{2}a^{7})(3^{2} + 2^{3}) |

(2^{2}a^{7})(2^{3}a^{2} + 3^{2}) | = |
---|---|

(2^{2}a^{7})(3^{2} + 2^{3}) |

2^{3}a^{2} + 3^{2} | = |
---|---|

3^{2} + 2^{3} |

8a^{2} + 9 |
---|

17 |

4. The answer is

32a^{9} + 36a^{7} | = |
---|---|

36a^{7} + 32a^{7} |

8a^{2} + 9 |
---|

17 |

The problem is solved.