Problem 39 - Algebra 2

Question

Simplify each expression. \(\frac{\left(-2 r s^{2}\right)^{2}}{12 r^{2} s^{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{3}s\).

Step 1: Expand the Numerator

First, expand the term in the numerator: \\((-2rs^2)^2 = (-2)^2r^2(s^2)^2 = 4r^2s^4\).\

Step 2: Write the Expression as a Fraction

Place the expanded form of the numerator over the original denominator: \\(\frac{4r^2s^4}{12r^2s^3}\).

Step 3: Cancel Out Common Factors

Divide both the numerator and the denominator by \(r^2\). This simplifies to: \\(\frac{4s^4}{12s^3}\).

Step 4: Simplify the Coefficients

Simplify the fraction \(\frac{4}{12}\) by dividing both the numerator and the denominator by their greatest common divisor, 4, resulting in: \\(\frac{1}{3}\).

Step 5: Simplify the Variable Part

For the \(s\) terms, subtract the exponent in the denominator from the exponent in the numerator: \\(s^{4-3} = s^1 = s\).

Step 6: Combine Simplified Terms

The simplified expression is the product of the simplified coefficient and variable: \\(\frac{1}{3}s\).

Key Concepts

Simplify

Simplifying rational expressions can feel like a puzzle, but it's all about finding and reducing the expression to its simplest form. When you simplify, your goal is to make the expression as easy to understand as possible. Consider each part of the expression individually:

• Numerator
• Denominator
• Any variables involved

Look first for common factors in both the numerator and the denominator. You can cancel these out, as they don't change the value of the expression. Once you reduce everything down, you've simplified the expression.

Exponent Rules

Exponent rules are a set of guidelines that make working with powers much more manageable. For instance, when you have a power raised to a power, like \((-2rs^2)^2\), you multiply the exponents. The result gives: \((-2)^2, r^2, s^{2 \times 2}\). Following these rules makes complex expressions less intimidating:

• Product of Powers: When multiplying, add exponents.
• Power of Power: Multiply the exponents.
• Quotient of Powers: Subtract exponents in division.

Being familiar with these simple rules allows you to handle any exponent-based operation with ease.

Fraction Reduction

Fraction reduction is about making a fraction as simple as possible. This means ensuring that the numerator and the denominator have no common factors other than 1. For example, simplifying \(rac{4}{12}\) involves dividing both by 4, their greatest common divisor, to get \(rac{1}{3}\). Implement these practices:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

When you reduce fractions, the values remain equivalent, but the expression becomes much clearer.

Algebraic Fractions

Algebraic fractions are just like numerical fractions but include variables, such as \(rac{r^2}{s^3}\). They represent a division between two algebraic expressions. To simplify these, treat variables just like numbers:

• Search for any common factors in variables across the numerator and the denominator.
• Apply exponent rules to simplify complex variables.

As you simplify algebraic fractions, cancel out any common terms, making sure to reduce to the least form. This helps in making complex equations easier to work with and solutions more apparent.