Problem 38 - Precalculus Mathematics for Calculus

Question

Perform the multiplication or division and simplify. $$\frac{x}{y / z}$$

Step-by-Step Solution

Answer

\( \frac{xz}{y} \) is the simplified form of the expression.

Step 1: Understand the Expression

The expression given is a fraction: \( \frac{x}{y/z} \). It involves division in the denominator, which simplifies to multiplying the numerator by the reciprocal of the denominator.

Step 2: Apply the Rule of Division of Fractions

When dividing by a fraction \( y/z \), you multiply by its reciprocal. The reciprocal of \( y/z \) is \( z/y \). Thus, the expression becomes \( x \times \frac{z}{y} \).

Step 3: Perform the Multiplication

Now, multiply the numerator \( x \) by the reciprocal \( \frac{z}{y} \). This results in: \( \frac{xz}{y} \).

Step 4: Simplify if Possible

Check if \( xz \) and \( y \) have any common factors that can be simplified. If there are no common factors, this is the simplest form.

Key Concepts

Fractions

Fractions represent a part of a whole and consist of a numerator (top number) and a denominator (bottom number). In algebra, fractions can involve variables in place of numbers. They are an essential concept to understand as they form the basis for many operations, especially in algebraic expressions.
To work effectively with fractions:

• Identify the numerator and denominator clearly.
• Perform operations like addition, subtraction, multiplication, or division according to rules specific to fractions.
• Understand that division by a fraction is the same as multiplication by its reciprocal.

Fractions often require simplification to express them in their simplest form. This is particularly important in algebra to ensure clarity and simplify further operations.

Reciprocal

The reciprocal of a fraction plays a crucial role in dividing fractions. The reciprocal is obtained by flipping the numerator and the denominator of the fraction. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
The reciprocal is useful because:

• It transforms division into multiplication, which is often easier to compute.
• It helps in simplifying expressions involving complex fractions.

When dividing fractions, like \( \frac{x}{y/z} \), you can easily convert division into multiplication by using the reciprocal of the divisor, turning the expression into multiplication, which is more straightforward to handle in algebraic expressions.

Simplification of Expressions

Simplification is the process of reducing an expression to its simplest form. It involves removing any unnecessary complexity without changing the expression's value. When simplifying, always look for opportunities to factor or cancel out common terms.
Key steps to simplify expressions include:

• Identify common factors in the numerator and denominator and cancel them out.
• Use algebraic identities and properties to transform expressions.
• In the context of fractions, simplifying may involve factoring polynomials in the numerator and denominator.

Simplification makes expressions easier to interpret and handle, especially in equations or when performing additional operations such as solving or graphing. It's a vital skill in algebra that reduces errors and assists in obtaining solutions in their cleanest form.