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Chapter 1: Problem 5

Perform the indicated divisions. $$\frac{8 x^{3} y^{2}}{-2 x y}$$

Short Answer

Expert verified
The simplified expression is \(-4x^2y\).

Step by step solution

01

Identify the Division

We are asked to simplify the expression \( \frac{8x^3y^2}{-2xy} \). This is a division of two monomials, where the numerator is \( 8x^3y^2 \) and the denominator is \( -2xy \).
02

Divide the Coefficients

Divide the numerical coefficients: \( \frac{8}{-2} = -4 \). So, the coefficient of the resulting expression is \(-4\).
03

Apply the Law of Exponents - Variable 'x'

Subtract the exponents of \( x \) in the numerator and denominator. We have \( x^3 \) in the numerator and \( x^1 \) (or \( x \)) in the denominator. The result is \( x^{3-1} = x^2 \).
04

Apply the Law of Exponents - Variable 'y'

Subtract the exponents of \( y \) in the numerator and denominator. We have \( y^2 \) in the numerator and \( y^1 \) (or \( y \)) in the denominator. The result is \( y^{2-1} = y^1 \) or simply \( y \).
05

Compose the Resulting Expression

Combine the results from the previous steps: the coefficient \(-4\), \( x^2 \), and \( y \). Thus, the simplified expression is \( -4x^2y \).

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Key Concepts

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

Monomial Division
Monomial division involves dividing one term by another, where each term consists of a constant (or coefficient) and variables raised to some powers. When performing monomial division, you divide both the coefficients and the variables separately. In the given expression \( \frac{8x^3y^2}{-2xy} \), two monomials are present. The numerator is \( 8x^3y^2 \) and the denominator is \( -2xy \).
  • Start by dividing the coefficients. In this case, divide \( 8 \) by \( -2 \), which gives you \( -4 \).
  • Next, deal with each variable. Divide the variables by subtracting the exponents of the same base.
This structured approach makes the division process straightforward, avoiding potential errors and ensuring clarity in your calculations.
Laws of Exponents
The laws of exponents are essential rules that simplify the manipulation of expressions involving powers. When dividing two expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Let's apply these laws to our expression:
  • For the variable \( x \), we have \( x^3 \) in the numerator and \( x^1 \) in the denominator. According to the laws of exponents, subtract the exponents: \( 3 - 1 = 2 \), resulting in \( x^2 \).
  • Similarly, for the variable \( y \), subtract the exponent in the denominator (\( y^1 \)) from the exponent in the numerator (\( y^2 \)), resulting in \( y^{2-1} = y \).
Understanding these rules simplifies complex expressions and helps avoid misuse of exponents, which could lead to mistakes in calculations.
Simplifying Expressions
Simplifying expressions transforms them into their simplest form, making them more understandable and often easier to work with. In our problem, after dividing both coefficients and using the laws of exponents to handle the variables, we arrive at the simplified form, \( -4x^2y \).
  • The coefficient \( -4 \) signifies that the resulting expression retains a negative value.
  • The variable \( x \) is now squared, meaning it has been reduced to \( x^2 \) through the division process.
  • The variable \( y \), simplified to \( y \), shows that it still plays a role but is no longer squared.
Throughout mathematics, simplifying expressions is crucial, transforming potentially complicated problems into solvable, manageable pieces. This process illuminates the path to accurate solutions, promoting confidence and clarity in problem-solving.

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Most popular questions from this chapter

Perform the indicated multiplications. The weekly revenue \(R\) (in dollars) of a flash drive manufacturer is given by \(R=x p,\) where \(x\) is the number of flash drives sold each week and \(p\) is the price (in dollars). If the price is given by the demand equation \(p=30-0.01 x,\) express the revenue in terms of \(x\) and simplify.

Find the indicated values. A formula used in determining the total transmitted power \(P_{t}\) in an AM radio signal is \(P_{t}=P_{c}\left(1+0.500 m^{2}\right) .\) Find \(P_{c}\) if \(P_{t}=680 \mathrm{W}\) and \(m=0.925\)

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One oil-well drilling rig drills \(100 \mathrm{m}\) deep the first day and \(200 \mathrm{m}\) deeper the second day. A second rig drills 200 m deep the first day and \(100 \mathrm{m}\) deeper the second day. In showing that the total depth drilled by each rig was the same, state what fundamental law of algebra is illustrated.

Perform the indicated operations and attach the correct units to your answers. $$\left(7.25 \frac{\mathrm{m}}{\mathrm{s}^{2}}\right)\left(\frac{1 \mathrm{ft}}{0.3048 \mathrm{m}}\right)\left(\frac{60 \mathrm{s}}{1 \mathrm{min}}\right)^{2}$$
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