Problem 15 - Intermediate Algebra

Question

Simplify. Write answers using positive exponents. \(\frac{4 x^{2} y^{3}}{8 x^{5} y^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y}{2x^{3}} \).

Step 1: Simplify the Coefficients

First, focus on simplifying the coefficients of the fraction. The coefficients are 4 in the numerator and 8 in the denominator. Simplify \( \frac{4}{8} \) by dividing both numerator and denominator by their greatest common divisor, which is 4. This gives us \( \frac{1}{2} \).

Step 2: Simplify the x-terms

Next, simplify the \( x \) terms. We have \( x^{2} \) in the numerator and \( x^{5} \) in the denominator. Apply the rule of exponents: \( \frac{x^{m}}{x^{n}} = x^{m-n} \). So, \( \frac{x^{2}}{x^{5}} = x^{2-5} = x^{-3} \).

Step 3: Simplify the y-terms

Now simplify the \( y \) terms. We have \( y^{3} \) in the numerator and \( y^{2} \) in the denominator. Using the rule \( \frac{y^{m}}{y^{n}} = y^{m-n} \), we get \( \frac{y^{3}}{y^{2}} = y^{3-2} = y^{1} = y \).

Step 4: Combine the Simplified Parts

Combine the simplified coefficients, \( x \), and \( y \) terms. The simplified expression is: \( \frac{1}{2} \cdot x^{-3} \cdot y = \frac{y}{2x^{3}} \), since having a negative exponent in the numerator is equivalent to making it a positive exponent in the denominator.

Key Concepts

Positive Exponents

Positive exponents are a fundamental concept in algebra. They indicate how many times a base is multiplied by itself. Consider the expression \(x^3\). Here, "\(x\)" is the base, and "3" is the exponent. This expression tells us to multiply \(x\) by itself three times: \(x \times x \times x\).
This is in contrast to negative exponents, which indicate division. An example is \(x^{-3}\), which is equal to \(\frac{1}{x^3}\). In problems like our example exercise, it is often necessary to rewrite negative exponents as positive ones to simplify the expression.

• Positive exponents mean repeated multiplication.
• Negative exponents imply division, or moving to the denominator.

Understanding and using positive exponents correctly helps in rewriting and simplifying more complex algebraic expressions.

Rules of Exponents

Rules of exponents simplify expressions involving powers. Here are some key rules:

• Product of Powers: When multiplying two powers that have the same base, add the exponents: \(x^m \cdot x^n = x^{m+n}\).
• Quotient of Powers: When dividing two powers with the same base, subtract the exponents: \(\frac{x^m}{x^n} = x^{m-n}\). This rule is crucial in our exercise, allowing us to simplify terms like \(\frac{x^2}{x^5}\) into \(x^{-3}\).
• Power of a Power: When raising a power to another power, multiply the exponents: \((x^m)^n = x^{m \cdot n}\).
• Zero Exponent Rule: Any base except zero raised to the power of zero is 1: \(x^0 = 1\).

These rules are essential for rewriting expressions so they only contain positive exponents, which makes them easier to handle in various mathematical operations.

Fraction Simplification

Simplifying fractions is important for reducing expressions to their simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). Let's look at the fraction \(\frac{4}{8}\):
1. The GCD of 4 and 8 is 4.
2. Divide both numbers by 4 to simplify: \(\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\).
This process is crucial not only for coefficients in fractions but also for algebraic expressions. Simplifying the coefficients comes first, making it easier to combine with simpler expressions of variables after using the rules of exponents. Using these steps ensures that any algebraic expression is at its most reduced and elegant form possible.