Problem 67 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(\frac{4 x^{3} y}{18 x^{2}} \cdot \frac{9}{16 x^{2} y}\) A. \(\frac{x}{16}\) B. \(\frac{x}{8}\) C. \(\frac{x^{3}}{8}\) D. \(\frac{1}{8 x}\)

Step-by-Step Solution

Answer

Therefore, the correct simplified version of the given expression is B. \( \frac{x}{8} \)

Step 1: Write out the Expression

Start by writing the expression from the problem: \( \frac{4 x^{3} y}{18 x^{2}} \cdot \frac{9}{16 x^{2} y} \)

Step 2: Break Down the Fractions

After writing out the expression, break down the expression into simplified fractions. This can be done by dividing each pair of similar terms (variables or constants) by each other if possible: \( \frac{4}{18} \cdot \frac{x^{3}}{x^{2}} \cdot \frac{y}{1} \cdot \frac{9}{16} \cdot \frac{1}{x^{2}} \cdot \frac{1}{y} \)

Step 3: Simplify Each Fraction

Now, simplify each fraction separately. To do so, reduce each fraction to the lowest terms possible. For the powers of x, subtract the exponent of the denominator from the exponent of the numerator since we're dividing: \( \frac{2}{9} \cdot x \cdot 1 \cdot \frac{9}{16} \cdot \frac{1}{x} \cdot 1 \)

Step 4: Simplify Further

Simplify the expression further taking care of multiplying constants, canceling out same variables and taking care of powers: \( \frac{2}{9} \cdot x \cdot \frac{9}{16} \cdot \frac{1}{x} = \frac{2 x 9}{9 x 16} = \frac{x}{8} \) since 2 and 16 simplify to 1 and 8, and x cancels out x. So, \( \frac{x}{8} \)

Step 5: Match with Options

Now, check the simplified expression with given alternatives: A. \( \frac{x}{16} \) B. \( \frac{x}{8} \) C. \( \frac{x^{3}}{8} \) D. \( \frac{1}{8x} \). The simplified expression matches with option B.

Key Concepts

Fractions in Algebra

Fractions in algebra are similar to number fractions, but they involve algebraic expressions. When dealing with algebraic fractions, rules for arithmetic operations such as addition, subtraction, multiplication, and division apply just like in numerical fractions. However, there's an additional step of simplifying these fractions by canceling out common terms.

In the given problem, start by identifying like terms in the numerators and denominators. Each term should be a product of coefficients and variables. Look for common factors. For example, in the expression \( \frac{4 x^{3} y}{18 x^{2}} \cdot \frac{9}{16 x^{2} y} \), notice how the coefficients 4 and 18 can be simplified to 2 and 9 by dividing both by 2 and 9 respectively.

• Cancel the common factors in the numerator and the denominator.
• Usually, the coefficients and variables are treated separately for simplification.
• Work step by step, ensuring all terms are accounted for.

By breaking down the whole expression into parts, the fractions were simplified easily to derive the answer \(\frac{x}{8}\). This systematic approach helps in getting a clear understanding of each step involved.

Exponents

Exponents are powerful tools in algebra that represent repeated multiplication. Understanding how they work is crucial to simplifying expressions involving powers. The expression \( \frac{x^3}{x^2} \cdot \frac{1}{x^2} \) involves several applications of exponent rules.

In general, when dividing terms with the same base, you subtract the exponents: \( x^{m} \div x^{n} = x^{m-n} \). This means that \( \frac{x^3}{x^2} \) simplifies to \( x^{3-2} = x^1 = x \). Applying this rule helps to simplify the given expression considerably.

• When no power is visible, it’s actually raised to the power of 1.
• Exponent rules particularly shine in simplifying terms and fractions involving variables.
• Remember that \( x^0 = 1 \), which can clear away terms completely.

Understanding these rules ensures that you handle the variables correctly and simplifies the solution process without errors.

Variables in Expressions

Variables are essential elements in algebra that represent unknown values. They behave like numbers in expressions, which means you can perform arithmetic operations on them. In expressions such as \( \frac{4 x^{3} y}{18 x^{2}} \cdot \frac{9}{16 x^{2} y} \), \( x \) and \( y \) are the variables.

Variables follow the same rules as other algebraic components in an expression. They can multiply, divide, and cancel out just like numbers when identical terms match in numerators and denominators.

• Same variable terms in numerators and denominators cancel out.
• Ensure careful handling of their powers using exponent rules.
• Organize expressions to group similar variables for easier simplification.

This is practical, especially for simplifying, and becomes second nature as you master handling them in various algebraic expressions. Through practice, recognizing patterns with variables will make solving algebraic problems seem logical and manageable.