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Chapter 5: Problem 21

For Problems \(1-24\), divide the monomials. $$ \frac{56 a^{2} b^{3} c^{5}}{4 a b c} $$

Short Answer

Expert verified
The result of the division is \( 14ab^2c^4 \).

Step by step solution

01

Simplify the Coefficient

Start by dividing the numerical coefficients of the monomials: divide 56 by 4. This gives 14 because \( \frac{56}{4} = 14 \).
02

Apply the Quotient Rule for Exponents

For each variable, use the rule \( \frac{x^m}{x^n} = x^{m-n} \). Apply this to each variable individually:- For \(a\): \( \frac{a^2}{a^1} = a^{2-1} = a^1 = a\).- For \(b\): \( \frac{b^3}{b^1} = b^{3-1} = b^2\).- For \(c\): \( \frac{c^5}{c^1} = c^{5-1} = c^4\).
03

Combine the Results

Combine the simplified coefficient with the simplified variable expressions. The final result is:\[ 14abc^4 \]

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Key Concepts

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

Quotient Rule for Exponents
When dividing monomials, one powerful tool at your disposal is the quotient rule for exponents. It comes in handy when you are working with expressions that involve the same base. This rule allows you to simplify these expressions more efficiently.
The quotient rule can be stated as follows: if you have two powers with the same base, such as \( x^m \) divided by \( x^n \), the result is \( x^{m-n} \). This means you subtract the exponent in the denominator from the exponent in the numerator.
  • It's a straightforward technique and particularly useful for dividing variables with exponents.
  • Always ensure that the bases are the same before applying this rule.
By consistently applying the quotient rule, you can simplify expressions step-by-step without getting overwhelmed by large numbers or complex polynomials.
Simplifying Monomials
Simplifying monomials involves reducing an expression to its simplest form by following established rules and operations like the quotient rule for exponents. In the given exercise, the monomial \( \frac{56 a^{2} b^{3} c^{5}}{4 a b c} \) involves simplifying terms containing both coefficients and variables.
Here鈥檚 a breakdown of the simplification process:
  • Start with the coefficients: divide 56 by 4 to get 14.

  • Apply the quotient rule to each variable separately:
    • For \( a^2 \) and \( a^1 \), subtract the exponents: \( a^{2-1} = a \).
    • For \( b^3 \) and \( b^1 \), it becomes \( b^{3-1} = b^2 \).
    • For \( c^5 \) and \( c^1 \), you reduce to \( c^{5-1} = c^4 \).
Reassembling these components gives you the simplified monomial: \( 14abc^4 \). Simplifying monomials is about careful application and attention to detail, ensuring each step is accurate.
Mathematical Problem Solving
Mathematical problem solving is both an art and a technique that involves intuition, logic, and methodical steps. Tackling algebraic problems, like the division of monomials, requires you to follow a sequence of actions.
The key to effective problem solving is breaking down a problem into smaller, manageable parts:
  • First, identify what you know, such as numerical coefficients and variables with exponents.

  • Second, apply relevant mathematical laws or rules, like the quotient rule for exponents.

  • Third, combine everything back together once simplification steps are completed, ensuring the solution is as straightforward as possible.
Alongside these steps, practice and experience are crucial. The more you engage with problems, the more skillful you become at predicting where errors might occur and how best to navigate through challenges. Mathematical problem solving is about patience and refinement, gradually honing your ability and understanding over time.

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Most popular questions from this chapter

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (4 a-7 b)^{2} $$

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (x+6 y)^{2} $$

Find the indicated products by using the shortcut pattern for multiplying binomials. $$ (y-7)(y-12) $$

For Problems \(82-112\), use one of the appropriate patterns \((a+b)^{2}=a^{2}+2 a b+b^{2},(a-b)^{2}=a^{2}-2 a b+b^{2}\), or \((a+b)(a-b)=a^{2}-b^{2}\) to find the indicated products. $$ (2 x+9)^{2} $$

For Problems \(113-120\), find the indicated products. Don't forget that \((x+2)^{3}\) means \((x+2)(x+2)(x+2)\). $$ (x-1)^{3} $$
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