Problem 64 - Algebra 2

Question

Simplify each expression. $$ \frac{9 x^{2} y^{3}}{(5 x y z)^{2}} \div \frac{(3 x y)^{3}}{20 x^{2} y} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{15xyz^2}\).

Step 1: Simplify the Exponents

We start by expanding the exponents in the numerator and denominator of each fraction separately. The problem is \[\frac{9x^2y^3}{(5xyz)^2}\div \frac{(3xy)^3}{20x^2y}.\]For the first fraction's denominator:\((5xyz)^2 = 5^2 \cdot x^2 \cdot y^2 \cdot z^2 = 25x^2y^2z^2\).For the second fraction's numerator:\((3xy)^3 = 3^3 \cdot x^3 \cdot y^3 = 27x^3y^3\).

Step 2: Rewrite the Division as Multiplication

Rewrite the division problem as multiplication by multiplying by the reciprocal of the second fraction:\[\frac{9x^2y^3}{25x^2y^2z^2} \times \frac{20x^2y}{27x^3y^3}.\]

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together:- Numerator: \(9x^2y^3 \times 20x^2y = 180x^4y^4\).- Denominator: \(25x^2y^2z^2 \times 27x^3y^3 = 675x^5y^5z^2\).This gives the new fraction:\[\frac{180x^4y^4}{675x^5y^5z^2}.\]

Step 4: Simplify the Fraction

We simplify the fraction by cancelling common factors:- Divide both the numerator and denominator by 45: \(\frac{180}{45} = 4\), and \(\frac{675}{45} = 15\).- Cancel common terms in the variables: - \(x^4/x^5 = 1/x\). - \(y^4/y^5 = 1/y\).Thus, the simplified form is:\[\frac{4}{15xyz^2}.\]

Key Concepts

Exponents Uncovered

Understanding exponents is crucial when dealing with algebraic fractions. Exponents are a shorthand way to express repeated multiplication of the same factor. For instance,

• \(x^2\) means \(x\) multiplied by itself: \(x \times x\).
• \(x^3\) means \(x \times x \times x\).

When simplifying expressions, especially those involving fractions and division, knowing how to handle exponents is key. Consider

• how you can expand expressions like \((5xyz)^2 = 5^2 \cdot x^2 \cdot y^2 \cdot z^2\).
• Similarly, for \((3xy)^3 = 3^3 \cdot x^3 \cdot y^3\).

Another important concept is power of a product, which allows us to expand expressions easily: the exponent outside affects every term inside the parenthesis, such as in \((ab)^n = a^n \cdot b^n\). Employing this knowledge helps simplify complex algebraic fractions by expanding each term and making it easier to identify common factors later on.

Simplifying Expressions with Precision

Simplifying expressions involves reducing an equation to its simplest form, making calculations and comparisons easier. In our given problem, we simplify by applying exponent rules and multiplying through fractions. It's important to first expand each exponent fully, as seen in prior steps. Once expanded, identify common factors you can cancel out. For simplification:

• Factorize numbers and variable terms within the fraction.
• Cancel anything identical on both the numerator and denominator, such as similar \(x\) and \(y\) terms.
• Simplify coefficients by finding a common divisor. For \(\frac{180}{675}\), divide both by 45 to get \(\frac{4}{15}\).

Identifying and cancelling out common terms is like pulling out matching blocks in a stack. They leave the simpler, remaining terms that are essential to the solution. Always complete this step to make sure the fraction is fully reduced. This results in a neat, easy-to-read and exact answer.

Art of Multiplying Fractions

Multiplying fractions involves a straightforward set of steps. The process is simpler than it seems since it only requires multiplying across the top (numerator) and the bottom (denominator). Consider two fractions,

• \(\frac{a}{b} \times \frac{c}{d}\).

Here's how you multiply:

• Numerators: Multiply \(a\) (from the first fraction) and \(c\) (from the second); this becomes your new top number.
• Denominators: Multiply \(b\) and \(d\); this gives the new bottom number.

For our problem, the multiplication step unified different variable terms, creating a combined expression. Always remember: unlike addition or subtraction, when multiplying fractions, keep everything straight across. This means respecting each term and multiplying once down the line without cross-contamination of terms. Finally, simplify where possible, ensuring the solution reflects cancellation or reduction of unnecessary clutter. Multiplying fractions is like crafting a simplified bridge, connecting complex components to form a cohesive and straightforward result.