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Chapter 6: Problem 91

Write each rational expression in lowest terms. $$ \frac{p+6}{p-6} $$

Short Answer

Expert verified
The rational expression \( \frac{p+6}{p-6} \) is already in its lowest terms.

Step by step solution

01

Identify the Rational Expression

The given rational expression is \( \frac{p+6}{p-6} \)
02

Factorize the Numerator and Denominator

Check if the numerator \( p+6 \) and the denominator \( p-6 \) can be factored further. In this case, both terms are already in their simplest forms.
03

Simplify the Rational Expression

Since the numerator \( p+6 \) and the denominator \( p-6 \) have no common factors other than 1, the rational expression \( \frac{p+6}{p-6} \) is already in its lowest terms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

numerator
In a rational expression, the numerator is the term above the fraction line. It's the top part of the fraction. For example, in the expression \( \frac{p+6}{p-6} \), the numerator is \( p+6 \). The numerator represents how many parts of the whole are being considered. To simplify a rational expression, check the numerator to see if it can be factored or reduced. This helps in further simplification. Also, remember that any operation performed to simplify must respect mathematical rules to maintain the expression's value.
denominator
The denominator is the term below the fraction line in a rational expression. It represents the total number of equal parts into which something is divided. In the expression \( \frac{p+6}{p-6} \), the denominator is \( p-6 \). To simplify, we need both the numerator and the denominator. Always check if the denominator can be factored or simplified. Keep in mind: the denominator must never be zero, as you can't divide by zero. Checking for common factors in the numerator and denominator helps find the simplest form.
lowest terms
Simplifying a rational expression means writing it in its lowest terms. This is the form where the numerator and denominator have no common factors other than 1. For \( \frac{p+6}{p-6}\), we checked and found no common factors, resulting in it being already at its simplest form. Steps for simplification include:
  • Factoring the numerator and denominator.
  • Finding common factors.
  • Dividing both terms by their greatest common factor (GCF).
Following these steps ensures your expression is as simple as possible.

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Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{z^{3}+27}{z^{3}-3 z^{2}+9 z} $$

Divide. Write each answer in lowest terms. $$ \frac{4 t^{4}}{2 t^{5}} \div \frac{(2 t)^{3}}{-6} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{(q-3)^{4}(q+2)}{q^{2}+3 q+2} \div \frac{q^{2}-6 q+9}{q^{2}+4 q+4} $$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{1}{2}+\frac{7}{8} $$

Divide. Write each answer in lowest terms. $$ \frac{8 z-16}{-20} \div \frac{3 z-6}{40} $$
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