Problem 15 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(x^{8} y^{6} \div\left(x^{3} y^{5}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(x^5 y\).

Step 1: Write Division as a Fraction

The expression \(x^{8} y^{6} \div (x^{3} y^{5})\) can be rewritten as a fraction: \(\frac{x^{8} y^{6}}{x^{3} y^{5}}\). This makes it easier to apply the rules of exponents.

Step 2: Simplify Each Base Separately

Apply the rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\). First, simplify the \(x\) terms: \(\frac{x^8}{x^3} = x^{8-3} = x^5\). Next, simplify the \(y\) terms: \(\frac{y^6}{y^5} = y^{6-5} = y^1\). As a result, \(y^1\) is just \(y\).

Step 3: Combine Simplified Expressions

Now that each base has been simplified, combine the results to form the final expression: \(x^5 y\).

Key Concepts

Rules of Exponents

Exponent rules make simplifying expressions involving powers much easier. Think of exponents as a shortcut for repeated multiplication. There are several key rules you should remember:

• Product of Powers Rule: If the bases are the same, you can simply add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Rule: If you are dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Zero Exponent Rule: Any base raised to the power of zero equals one: \(a^0 = 1\), where \(a eq 0\).

Use these rules whenever you deal with exponents to simplify calculations and solve problems quickly. They are extremely handy in algebra and beyond, making complex expressions more manageable.

Division of Monomials

Monomials are single-term algebraic expressions consisting of a coefficient and variables raised to powers. When dividing monomials, it’s crucial to apply the rules of exponents to simplify the expression effectively. Here are the steps to do so:1. **Rewrite the Division as a Fraction:** This step helps visualize the terms more clearly. For example, \(x^8 y^6 \div (x^3 y^5)\) can be rewritten as \(\frac{x^8 y^6}{x^3 y^5}\).2. **Apply the Quotient of Powers Rule:** For each variable with the same base, subtract the exponents. In our example: - For the \(x\) terms: \(\frac{x^8}{x^3} = x^{8-3} = x^5\). - For the \(y\) terms: \(\frac{y^6}{y^5} = y^{6-5} = y^1\), which simplifies to just \(y\).This process helps break down or reduce the complexity of the expression, making it straightforward and simpler to handle mathematically.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing fractions to their simplest form, just like numerical fractions. Here are the steps:1. **Identify Common Terms:** Look for common bases in the numerator and denominator that can be simplified using exponent rules.2. **Apply Exponent Rules:** Use the quotient of powers rule to simplify each variable separately. Subtract the exponents of the like bases accordingly.3. **Combine Simplified Terms:** After simplifying the variables, combine them to get the final expression.For instance, starting with \(\frac{x^8 y^6}{x^3 y^5}\), applying the rules leads to \(x^5 y\). Each step ensures the fraction is as reduced as possible, omitting any unnecessary terms or expressions. This simplification makes further manipulation or integration into larger equations easier, highlighting the importance of mastering this technique early in your studies.