Problem 74 - Algebra for College Students

Question

Perform the indicated operations. $$\frac{\left(-w y^{2}\right)^{3}}{3 w^{2} y} \cdot \frac{(2 w y)^{2}}{4 w y^{3}}$$

Step-by-Step Solution

Answer

\( \frac{-w^{2} y^{4}}{3} \)

Step 1: Simplify the Numerators

First simplify the numerators of each fraction.The first numerator is \((-w y^{2})^{3}\). Raising each term inside the parenthesis to the third power, we get: \((-w y^{2})^{3} = (-w)^{3} \times (y^{2})^{3} = -w^{3} y^{6}\).The second numerator is \((2wy)^{2}\). Raising each term inside the parenthesis to the second power, we get: \((2wy)^{2} = 2^{2} \times w^{2} \times y^{2} = 4w^{2}y^{2}\).

Step 2: Simplify the Denominators

Next, simplify the denominators of each fraction.The first denominator is \(3 w^2 y\), which is already simplified.The second denominator is \(4 w y^{3}\), which is also simplified.

Step 3: Rewrite the Expression

Rewrite the expression using the simplified numerators and denominators: \(\frac{-w^{3} y^{6}}{3 w^{2} y} \times \frac{4 w^{2} y^{2}}{4 w y^{3}} \)

Step 4: Combine the Fractions

Combine the fractions into one, multiplying the numerators together and the denominators together: \(\frac{-w^{3} y^{6} \times 4 w^{2} y^{2}}{3 w^{2} y \times 4 w y^{3}} = \frac{-4 w^{5} y^{8}}{12 w^{3} y^{4}}\)

Step 5: Simplify the Result

Simplify the combined fraction by canceling out common factors in the numerator and the denominator.Divide the numerator and the denominator by \(4 w^3 y^4\) \(\frac{-4 w^{5} y^{8}}{12 w^{3} y^{4}} = \frac{-w^{2} y^{4}}{3}\)

Key Concepts

exponents

When working with exponents, you are essentially dealing with repeated multiplication of a number. For example, in the expression \((-w y^{2})^{3}\), the exponent 3 tells us to multiply \(-w y^{2}\) by itself three times. When raising a term with an exponent to another exponent, you multiply the exponents: \( (y^{2})^{3} = y^{6}\). This can also apply to negative bases and multiple variables within the same parenthesis.

multiplying fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. In our exercise, we had two fractions: \(\frac{\text{-w}^{3} y^{6}}{3 w^{2} y}\) and \(\frac{4 w^{2} y^{2}}{4 w y^{3}}\). To multiply them, combine the numerators and then combine the denominators: \(\frac{\text{-4 w}^{5} y^{8}}{12 w^{3} y^{4}}\). It's strategic to simplify each fraction as much as possible before multiplying to make the process easier.

simplifying expressions

Simplifying expressions is very valuable in algebra. It often involves distributing, combining like terms, or reducing fractions. In our solution, we simplified the fractions before multiplying and then simplified the final product. We divided the numerator and the denominator by common factors: \(\frac{-4 w^{5} y^{8}}{12 w^{3} y^{4}} = \frac{-w^{2} y^{4}}{3}\). By doing this, we ended up with the simplest form of the expression.

algebraic operations

Algebraic operations include addition, subtraction, multiplication, and division of algebraic expressions. In our example, the main operations were multiplication and division. We used these operations to combine and simplify the fractions correctly: from raising each term by an exponent to multiplying fractions and then dividing to cancel out common factors. These fundamental skills build a foundation for solving more complex algebraic problems.