Problem 62 - Algebra for College Students

Question

Find each quotient. $$\frac{56 x^{6} y^{4}}{-7 x^{2} y^{3}}$$

Step-by-Step Solution

Answer

The quotient is \(-8x^4y\).

Step 1: Simplify the numerical coefficients

Divide the coefficients 56 and -7. The quotient of 56 and -7 is -8, since \( \frac{56}{-7} = -8 \).

Step 2: Simplify the powers of x

Apply the quotient rule for exponents: \( \frac{x^a}{x^b} = x^{a-b} \). Here, \( x^6 \) divided by \( x^2 \) results in \( x^{6-2} = x^4 \).

Step 3: Simplify the powers of y

Similarly, for \( y \), \( \frac{y^4}{y^3} = y^{4-3} = y^1 = y \).

Step 4: Combine the simplified components

Combine the results from Steps 1-3: The quotient is \( -8x^4y \).

Key Concepts

Quotient Rule for Exponents

In algebra, the quotient rule for exponents is a fundamental concept used to simplify expressions involving division of like bases. When you divide two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This rule is expressed mathematically as \( \frac{x^a}{x^b} = x^{a-b} \).
For example, if you have \( \frac{x^6}{x^2} \), you apply the quotient rule by subtracting 2 from 6, resulting in \( x^{6-2} = x^4 \). This rule helps in reducing the complexity of expressions and is widely used in algebra and calculus.

• It only applies to terms with the same base.
• Always subtract the lower exponent from the higher exponent.

Using this rule simplifies the process of handling large algebraic expressions, making problem-solving more efficient.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing an expression to its simplest form while maintaining its original value. This process often means reducing fractions, combining like terms, and applying exponent rules. It's like untangling a complex knot of numbers and variables.
Let's say you have a fraction \( \frac{56x^6y^4}{-7x^2y^3} \). To simplify, you first handle the numerical coefficients. Divide 56 by -7 to get -8. Next, apply the quotient rule separately to the variables: subtract the exponents of like bases. For \( x^6 \) and \( x^2 \), you get \( x^{6-2} = x^4 \), and for \( y^4 \) and \( y^3 \), you get \( y^{4-3} = y \).
By combining these steps, the simplified expression is \( -8x^4y \). This approach allows you to manage complex expressions effectively and is vital in solving equations accurately.

• Simplification maintains the value of expressions.
• It involves processes like factoring, canceling terms, and applying algebraic rules.

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, similar to long division with numbers. While the original problem does not involve dividing polynomials of multiple terms, understanding polynomial division can help when faced with more complex algebraic expressions.
Typically, you align the terms of both the dividend and the divisor in descending order of their exponents. Divide the leading term of the dividend by the leading term of the divisor, then multiply the entire divisor by that result and subtract from the original polynomial.
Though the process can be tedious, it's straightforward:

• Repeat the divide-multiply-subtract cycle until you can't anymore.
• The quotient obtained can be a simpler expression.

This method is essential for tasks like finding limits, solving equations, and integrating complex polynomials in calculus. Understanding the basics of polynomial division builds a strong foundation in algebraic proficiency.