Problem 59 - Algebra for College Students

Question

Find each quotient. $$\frac{9 x^{4} y^{5}}{3 x y^{2}}$$

Step-by-Step Solution

Answer

The quotient is \(3x^3y^3\).

Step 1: Separate Numerator and Denominator

Identify the numerator as \(9x^4y^5\) and the denominator as \(3xy^2\). The goal is to simplify the expression by dividing coefficients and subtracting the exponents of like bases.

Step 2: Divide Coefficients

Divide the numerical coefficients. \(9 \div 3 = 3\). So, the coefficient of the quotient is 3.

Step 3: Subtract Exponents of x

Apply the rule \(x^m \div x^n = x^{m-n}\). For \(x\), subtract the exponent in the denominator from the exponent in the numerator: \(4 - 1 = 3\). So, the term becomes \(x^3\).

Step 4: Subtract Exponents of y

Similarly, apply the rule \(y^m \div y^n = y^{m-n}\). For \(y\), subtract the exponent in the denominator from the exponent in the numerator: \(5 - 2 = 3\). So, the term becomes \(y^3\).

Step 5: Write Final Simplified Expression

Combine the simplified terms: the coefficient 3, the power of \(x\) as \(x^3\), and the power of \(y\) as \(y^3\). The final simplified expression is \(3x^3y^3\).

Key Concepts

Quotients

In algebra, a quotient is the result of dividing one expression by another. It's similar to division in arithmetic but involves elements with variables, such as polynomials or monomials. To find the quotient of two algebraic expressions, we often separate the task like a recipe:

• Identify the numerator and denominator: First, spot the numbers and variables in the expression. For example, if given \( \frac{9x^4y^5}{3xy^2} \), identify the top part (numerator) \( 9x^4y^5 \), and the bottom part (denominator) \( 3xy^2 \).
• Divide coefficients: Compute by dividing the numbers directly like you would in simple arithmetic. Here, it’s \( 9 \div 3 = 3 \).
• Simplify the variables: Apply rules of division specific to expressions. This means subtracting the exponents of matching variables from the numerator and the denominator, like \( x^4 \div x = x^3 \).

Thus, mastering the concept of quotients allows you to look beyond the numbers and simplify complex expressions into manageable pieces.

Simplifying Expressions

Simplifying expressions brings order from complexity. It involves reducing algebraic expressions to their simplest form, where no further operations can be performed. This simplification process makes expressions easier to understand, graph, or solve.To simplify an expression such as \( 9x^4y^5 \div 3xy^2 \), follow these steps:

• Divide the numbers: Break down the coefficients by division, \( 9 \div 3 \), yielding a smaller number, \( 3 \).
• Reduce variable parts: For each like variable, employ exponent laws to subtract the smaller exponent from the larger, simplifying terms like \( y^5 \div y^2 = y^3 \).

This tidying-up approach changes large, unwieldy expressions into refined versions, facilitating further calculations like graphing or solving equations.

Exponents

Exponents appear as small numbers positioned to the top right of a base number; they signify repeated multiplication. For example, \( x^4 \) means "\( x \) multiplied by itself four times." When dividing terms with exponents, a simple rule reigns: subtract the exponent in the denominator from the exponent in the numerator.For instance:

• Same base: Use \( a^m \div a^n = a^{m-n} \). So, \( x^4 \div x = x^{4-1} = x^3 \).
• Multiple bases: Apply the rule to every similar base, reducing each variable component separately.

Understanding and applying exponent rules are essential to handling and simplifying expressions in algebra.

Polynomials

Polynomials are expressions composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. They can range from simple monomials like \( x \) to more complicated terms such as \( 9x^4y^5 \).Key characteristics:

• Terms: Individual parts of a polynomial separated by addition or subtraction. Each term includes a coefficient (a number like \( 9 \)) multiplied by variables raised to an exponent (\( x^4 \)).
• Simplification: Involves reducing terms using rules of operations and exponents.
• Division: Polynomials can be divided by other polynomials or monomials using strategies similar to dividing numerical fractions.

Becoming familiar with polynomials removes the intimidation of algebra, unveiling a structured yet flexible world of numeric and variable interactions.