What are the rules for simplifying fractions with negative exponents?

Rules for simplifying fractions with negative exponents:

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 negative exponents

8a^{-6} |
---|

12a^{-4} |

Solution.

Step by step.

1. Factor the numerator and denominator of the fraction.

The prime factorization of 8

2 * 2 * 2 =

2^{3}

2

The prime factorization of 12

2 * 2 * 3 =

2^{2} * 3

2

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

We have

8a^{-6} | = |
---|---|

12a^{-4} |

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

2^{2} * a^{-4} * 3 |

The common factors are 2^{2} and a^{-4}.

3. Cancel down all the common factors.

Cancel the common factors

(2^{2} * a^{-4}) * 2 * a^{-2} | = |
---|---|

(2^{2} * a^{-4}) * 3 |

(2^{2} * a^{-4}) * 2 * a^{-2} | = |
---|---|

(2^{2} * a^{-4}) * 3 |

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

3 |

4. The answer is

8a^{-6} | = | 2a^{-2} |
---|---|---|

12a^{-4} | 3 |

The problem is solved.