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Chapter 10: Problem 586

Divide the monomials. $$\frac{28 a^{9} b}{7 a^{4} b^{3}}$$

Short Answer

Expert verified
\[ \frac{4a^5}{b^2} \]

Step by step solution

01

- Simplify the coefficients

Divide the numerical coefficients. In this case, divide 28 by 7: \[ \frac{28}{7} = 4.\] So the expression simplifies to \[ 4 \frac{a^9 b}{a^4 b^3}.\]
02

- Simplify the variable with the higher exponent (a)

Subtract the exponents of the variable \(a\) since it appears in both the numerator and denominator: \[ a^{9-4} = a^5.\] The expression simplifies to \[ 4 a^5 \frac{b}{b^3}.\]
03

- Simplify the variable with the higher exponent (b)

Subtract the exponents of the variable \(b\) since it appears in both the numerator and denominator: \[ b^{1-3} = b^{-2}.\] The expression simplifies to \[ 4 a^5 b^{-2}.\]
04

- Rewrite with positive exponents

Rewrite the expression with positive exponents by using the property \( b^{-2} = \frac{1}{b^2} \). The final simplified expression is \[ 4 a^5 \frac{1}{b^2} = \frac{4 a^5}{b^2}. \]

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Key Concepts

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

Simplifying Exponents
One crucial step in dividing monomials is simplifying the exponents. Here, we manage the exponents separately from the coefficients. Simplifying exponents often involves subtraction. Since we're dividing, we subtract the exponent in the denominator from the exponent in the numerator for each variable. For example, if we have \(a^9\) in the numerator and \(a^4\) in the denominator, we subtract 4 from 9. This leaves us with \(a^{9-4} = a^5\). Simplifying exponents ensures that our expressions are easier to handle and understand.
Polynomial Division
Dividing monomials is a specific type of polynomial division. We separate the coefficients and the variables, handling them individually. Follow these steps easily:
  • Divide the coefficients: \(\frac{28}{7} = 4\).
  • Simplify the exponents for each variable by subtracting the exponent in the denominator from the exponent in the numerator. For example, \(a^9 \div a^4 = a^{9-4} = a^5\).
  • Combine these results: \(4a^5\).
    • You can always follow this systematic approach for any similar problem.
Remember, organizing your work step-by-step simplifies the process and reduces errors.
Positive Exponents
After simplifying, we should express any negative exponents as positive exponents for the final answer. This is a standard practice in mathematics, ensuring clarity and consistency. For example, if we get an expression \(b^{-2}\), converting it to a positive exponent gives us \(\frac{1}{b^2}\). This approach makes the expressions cleaner and easier to read. For instance, in our solution:
  • We obtained \(4a^5 b^{-2}\).
  • We then rewrite \(b^{-2}\) as \(\frac{1}{b^2}\).
  • Combining it all, we get \(\frac{4a^5}{b^2}\).
    • Always rewrite negative exponents positively for better readability and understanding.

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Most popular questions from this chapter

Simplify each expression. $$\left(2.4 \times 10^{8}\right)\left(2 \times 10^{-5}\right)$$

Multiply the binomials using various methods. $$(x-8)(x+9)$$

Simplify. $$(-10)^{-3}$$

Convert \(5.25 \times 10^{-4}\) to decimal form.

Factor the greatest common factor from each polynomial. $$6 p^{2}+6 p$$
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