Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\frac{x}{yz}\)

Step 1: Understand the Expression

The given expression is \(\frac{x / y}{z}\), which we need to simplify. Notice that this is a division of fractions. The expression \(x / y\) is divided by \(z\).

Step 2: Express Division as Multiplication

To simplify the expression \(\frac{x / y}{z}\), convert the division into multiplication by using the reciprocal. The division by \(z\) is the same as multiplying by \(\frac{1}{z}\). Thus, the expression becomes \((x / y) \times \frac{1}{z}\).

Step 3: Multiply the Fractions

Now, multiply the fractions: \(\frac{x}{y} \times \frac{1}{z}\). This results in \(\frac{x \times 1}{y \times z} = \frac{x}{yz}\).

Step 4: Simplify the Result

The expression \(\frac{x}{yz}\) is already as simplified as possible unless there are common factors between \(x\), \(y\), and \(z\). Since they are different variables, the simplified form is \(\frac{x}{yz}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is about reducing them to their most straightforward form. In fraction terms, simplifying means finding an equivalent fraction where the numerator and the denominator are as small as possible. This involves dividing both by their greatest common divisor (GCD).
To simplify:

• Check if the numbers in the fraction share any common factors.
• If they do, divide both the numerator and the denominator by the highest common factor.
• The result will be the simplest form of the fraction.

Remember, if your variables represent numbers with specific factors, simplifying is the best way to present your fraction as clean and neat. In our exercise, \(rac{x}{yz}\) is already simplified, given x, y, and z are distinct variables without shared factors.

Multiplication of Fractions

Multiplying fractions is simpler than it might seem. The key is to remember to multiply across both the numerators and the denominators.
A step-by-step guide to multiplying fractions:

• Multiply the numerators to obtain the new numerator.
• Multiply the denominators to obtain the new denominator.
• If possible, simplify the resulting fraction.

In our example, multiplying \(rac{x}{y}\) by \(rac{1}{z}\) gives \(rac{x imes 1}{y imes z}\), which equals \(rac{x}{yz}\). The multiplication results in a new fraction, and the simplicity lies in working straight across each fraction part.

Division of Fractions

Dividing fractions might seem tricky, but it becomes easy when you know the rule: "Multiply by the reciprocal." When you divide by a fraction, you flip it over and multiply.
To divide fractions:

• Take the reciprocal (or invert) of the divisor—the fraction you are dividing by.
• Change the operation from division to multiplication.
• Follow the rules for multiplying fractions.

In our exercise, \(rac{x / y}{z}\) transforms to multiplication: \((x / y) \times \frac{1}{z}\), leading to \(\frac{x}{yz}\). The division by \(z\) turns into multiplying by \(\frac{1}{z}\), simplifying the problem and easing the calculation.