Problem 85 - Algebra for College Students

Question

Perform the indicated operations. Variables in exponents represent integers. $$\frac{x^{a}}{y^{2}} \cdot \frac{y^{b+2}}{x^{2 a}}$$

Step-by-Step Solution

Answer

\( \frac{y^{b}}{x^{a}} \)

Step 1: Write the Original Expression

Start with the given expression: the fraction you're given is: the expression is: the fraction is: \[ \frac{x^{a}}{y^{2}} \cdot \frac{y^{b+2}}{x^{2a}} \]

Step 2: Combine the Numerators

Combine the numerators into a single fraction: \[ x^{a} \times y^{b+2} \]

Step 3: Combine the Denominators

Combine the denominators into a single fraction: \[ y^{2} \times x^{2a} \]

Step 4: Simplify the Fraction

Simplify the resultant fraction by combining the terms with the same base: \[ \frac{x^{a} \times y^{b+2}}{x^{2a} \times y^{2}} \]

Step 5: Simplify the Expression

Apply the division properties of exponents: \[ \frac{x^{a}}{x^{2a}} \times \frac{y^{b+2}}{y^{2}} = x^{a-2a} \times y^{(b+2)-2} = x^{-a} \times y^{b} \]

Step 6: Final Simplified Expression

Combine the simplified terms to get the final result: \[ x^{-a} \times y^{b} = \frac{y^{b}}{x^{a}} \]

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial in simplifying exponential expressions. An exponent tells how many times we multiply a number by itself. For example, in the expression \(x^a\), \(x\) is the base, and \(a\) is the exponent. Here are some important properties of exponents you should know:

• Product of Powers Property: If you are multiplying two expressions with the same base, you simply add their exponents: \(x^a \times x^b = x^{a+b}\).
• Quotient of Powers Property: When dividing two expressions with the same base, subtract the exponents: \(\frac{x^a}{x^b} = x^{a-b}\).
• Power of a Power Property: If you have an exponent raised to another exponent, multiply the exponents: \((x^a)^b = x^{a \times b}\).
• Zero Exponent Rule: Any number raised to the power of zero equals 1: \(x^0 = 1\), as long as \(x eq 0\).
• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base: \(x^{-a} = \frac{1}{x^a}\).

By understanding these properties, it becomes easier to handle different algebraic expressions involving exponents.

Fraction Simplification

Simplifying fractions, especially those involving exponents, follows a systematic process. Let's look at how you can simplify fractions step-by-step:

• Combine Numerators and Denominators: In an expression like \(\frac{x^a}{y^2} \cdot \frac{y^{b+2}}{x^{2a}}\), the first step is to rewrite it as a single fraction: \(\frac{x^a \times y^{b+2}}{y^2 \times x^{2a}}\).
• Simplify Powers: Use the properties of exponents to combine and simplify terms. For the numerator, you have \(x^a \times y^{b+2}\), and for the denominator, \(x^{2a} \times y^2\).
• Cancel Out Common Factors: Apply the quotient of powers property: \(\frac{x^a}{x^{2a}} = x^{a-2a} = x^{-a}\). Similarly, for \(\frac{y^{b+2}}{y^2} = y^{(b+2)-2} = y^b\).
• Express in Simplified Form: Your final, simplified form should not have any common factors in the numerator and denominator: \(x^{-a} \times y^b = \frac{y^b}{x^a}\).

By following these steps, you can more easily work through complex fractions.

Algebraic Expressions

Algebraic expressions can be intimidating, but they become manageable once you break them down. Here’s how you can approach them:

• Identify the Variables and Constants: Look at the variables (like \(x\) and \(y\)) and constants (numerical values) involved.
• Focus on Exponents: Understand how to handle variables with exponents, keeping in mind the properties we've discussed.
• Write the Expression Clearly: For example, the given exercise involves: \(\frac{x^a}{y^2} \cdot \frac{y^{b+2}}{x^{2a}}\).
• Combine and Simplify Terms: Use the properties of exponents to combine terms with common bases, then simplify using fraction rules.
• Simplify Step-by-Step: As seen in the solution steps, combining the numerators and denominators first, then simplifying exponents, eventually leads to \(\frac{y^b}{x^a}\).

By breaking down each step, any seemingly difficult algebraic expression can be simplified into a more understandable form. Always take it one step at a time to ensure accuracy.