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Chapter 6: Problem 417

Divide the monomials. $$\frac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$$

Short Answer

Expert verified
\[\frac{65a^{3} b^{2}}{42 c^{3}} \]

Step by step solution

01

- Divide the Coefficients

Divide the numerical coefficients of the terms. \[\frac{65}{42} \]
02

- Apply the Quotient Rule for the Variables

Subtract the exponents of the same bases from the numerator and the denominator: \[\frac{a^{10}}{a^{7}} = a^{10-7} = a^{3} \frac{b^{8}}{b^{6}} = b^{8-6} = b^{2} \frac{c^{5}}{c^{8}} = c^{5-8} = c^{-3} \]
03

- Combine the Results

Combine the results from the previous steps: \[\frac{65}{42} \times a^{3} \times b^{2} \times c^{-3} \]
04

- Simplify and Handle Negative Exponents

Rewrite any term with a negative exponent as a fraction: \[\frac{65}{42} \times a^{3} \times b^{2} \times \frac{1}{c^{3}} \] Thus, you get: \[\frac{65a^{3} b^{2}}{42 c^{3}} \]

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Key Concepts

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

Monomials
Monomials are algebraic expressions that consist of only one term. They can include numbers, variables, or the product of both.

For example, in the given exercise, both the numerator and the denominator are monomials:

  • The numerator is ewline
    • 65 \(a^{10} b^{8} c^{5}\).
  • The denominator is ewline
    • 42 \(a^{7} b^{6} c^{8}\).
Understanding monomials is the basic foundation for many algebraic operations. They help simplify complex expressions and make calculations easier.
Quotient Rule
The Quotient Rule is essential for dividing monomials. It states that to divide like bases, you subtract the exponents in the denominator from the exponents in the numerator.

Here's how we applied it in the solution:

  • For \(a\): ewline
    • \(\frac{a^{10}}{a^{7}} = a^{10-7} = a^{3}\).
  • For \(b\): ewline
    • \(\frac{b^{8}}{b^{6}} = b^{8-6} = b^{2}\).
  • For \(c\): ewline
    • \(\frac{c^{5}}{c^{8}} = c^{5-8} = c^{-3}\).
By using the quotient rule, we can easily manage the division of variables with exponents.
Negative Exponents
Negative exponents occur in algebra when you subtract larger exponents in the denominator from smaller exponents in the numerator. The result is a negative exponent.

For instance, in our exercise, we obtained a negative exponent for variable \(c\):

  • \(c^{5-8} = c^{-3}\).
Negative exponents can be simplified using the rule:
  • \(x^{-n} = \frac{1}{x^n}\).
This turns the negative exponent into a positive exponent in the denominator:

  • \(c^{-3} = \frac{1}{c^{3}}\).
This simplification is crucial for presenting a cleaner, more manageable final answer.
Simplification
Simplification is the process of making an algebraic expression as simple as possible.

In our exercise, the simplification steps were:

  • Divide the coefficients:ewline
    • \(\frac{65}{42}\).
  • Apply the quotient rule to each variable:ewline
    • \(a^{3}, b^{2}, c^{-3}\).
  • Handle negative exponents:ewline
    • \(c^{-3} = \frac{1}{c^{3}}\).
  • Combine all parts into one expression:ewline
    • \(\frac{65a^{3} b^{2}}{42 c^{3}}\).
By breaking down the steps and following the rules of algebra, we achieve a simplified, clear answer.

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Most popular questions from this chapter

Mixed Practice (a) \(q^{5} \cdot q^{3}\) (b) \(\left(q^{5}\right)^{3}\)

Divide each polynomial by the monomial. $$\frac{49 c^{2} d^{2}-70 c^{3} d^{3}-35 c^{2} d^{4}}{7 c d^{2}}$$

Divide each polynomial by the monomial. $$\left(9 n^{4}+6 n^{3}\right) \div 3 n$$

Simplify. (a) \(5^{-3}\) (b) \(10^{-5}\)

Simplify. (a) \(-5^{-3}\) (b) \(\left(-\frac{1}{5}\right)^{-3}\) |(c) \(-\left(\frac{1}{5}\right)^{-3}\) (d) \((-5)^{-3}\)
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