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Chapter 6: Problem 60

Divide and, if possible, simplify. $$ \frac{3 a+15}{a^{9}} \div \frac{a+5}{a^{8}} $$

Short Answer

Expert verified
\( \frac{3}{a} \)

Step by step solution

01

- Write the division as multiplication

Dividing by a fraction is equivalent to multiplying by its reciprocal. Rewrite the expression: equation \[\frac{\frac{3a + 15}{a^9}}{\frac{a + 5}{a^8}} = \frac{3a + 15}{a^9} \times \frac{a^8}{a + 5}\]
02

- Simplify the expression

Now, simplify the expression. Notice that \( 3a + 15 \) can be factored as \( 3(a + 5) \). Substitute and simplify: \[\frac{3(a + 5)}{a^9} \times \frac{a^8}{a + 5}\]
03

- Cancel common factors

Cancel the common factors, \( a + 5 \): \[= \frac{3 \cancel{(a + 5)}}{a^9} \times \frac{a^8}{\cancel{(a + 5)}} = \frac{3}{a^9} \times a^8\]
04

- Combine and simplify

Combine the remaining expressions. When multiplying, add the exponents of the same base: \[\frac{3}{a^9} \times a^8 = 3 \times a^{8-9} = 3 \times a^{-1} = \frac{3}{a}\]

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Key Concepts

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

Fraction Division
When dealing with fraction division, it's important to understand a simple rule: dividing by a fraction is the same as multiplying by its reciprocal. For example, dividing by \(\frac{a+5}{a^8}\) is equivalent to multiplying by its reciprocal \(\frac{a^8}{a+5}\).
This approach makes complex fraction problems easier to handle by converting them into multiplication problems. Always start by rewriting the division as a multiplication to simplify further steps.
Reciprocal
The reciprocal of a fraction is simply flipping the numerator and denominator. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
In algebraic terms, taking the reciprocal is just as important. When you see a fraction division problem, first find the reciprocal of the fraction you're dividing by. This step is crucial when simplifying complex algebraic expressions.
Simplification
Simplifying an algebraic expression means to rewrite it in its simplest or most condensed form.
This often involves:
  • Factoring expressions
  • Cancelling out common factors
For instance, in the given exercise, notice that \(3a + 15\) can be factored to \(3(a + 5)\).
By recognizing these patterns, you can cancel out factors and simplify the expression considerably. Always look for common factors and try to reduce the terms step by step.
Exponent Rules
Understanding exponent rules is fundamental in simplifying algebraic expressions. Some key rules include:
  • Multiplying powers: \(a^m \times a^n = a^{m+n}\)
  • Dividing powers: \(a^m / a^n = a^{m-n}\)
  • Power of a power: \((a^m)^n = a^{m \times n}\)
For example, in the exercise, the exponents are combined using the rule \(a^{8} / a^{9} = a^{8-9} = a^{-1}= \frac{1}{a}\).
With these rules in mind, you'll be able to handle even the toughest algebraic expressions confidently.

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Most popular questions from this chapter

If \(y\) varies inversely as the cube of \(x\) and \(x\) is multiplied by \(0.5,\) what is the effect on \(y ?\)

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For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\) $$ \begin{array}{l}{f(x)=\frac{2 x+5}{x^{2}+4 x+3}} \\\ {g(x)=\frac{x+2}{x^{2}-9}+\frac{x-1}{x^{2}-2 x-3}}\end{array} $$
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