Problem 76 - Elementary Algebra

Question

\(\left(\frac{3 x^{4}}{5 z^{8}}\right)\left(\frac{5 z^{8}}{3 x^{4}}\right)\)

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1 - Identify the Problem

The given expression is \(\frac{3 x^{4}}{5 z^{8}} \times \frac{5 z^{8}}{3 x^{4}}\). This is a multiplication of two fractions.

Step 2 - Apply the Property of Multiplication

When multiplying two fractions, multiply the numerators together and the denominators together: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).

Step 3 - Multiply the Numerators

Multiply the numerators of each fraction: \((3 x^{4}) \times (5 z^{8}) = 15 x^{4} z^{8}\).

Step 4 - Multiply the Denominators

Multiply the denominators of each fraction: \((5 z^{8}) \times (3 x^{4}) = 15 z^{8} x^{4}\).

Step 5 - Simplify the Expression

Combine the numerator and denominator: \(\frac{15 x^{4} z^{8}}{15 z^{8} x^{4}}\).

Step 6 - Cancel Common Factors

Since the numerator and denominator are identical, simplify by canceling out the common factors: \(\frac{15 x^{4} z^{8}}{15 z^{8} x^{4}} = 1\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a vital skill in algebra and helps in making calculations easier. When we simplify a fraction, we reduce it to its simplest form by removing any common factors that the numerator and the denominator share. For example, let's say we have the fraction \(\frac{8}{12}\). Both the numerator (8) and the denominator (12) can be divided by their greatest common divisor (GCD), which is 4:

• Divide the numerator by 4: \(\frac{8}{4} = 2\)
• Divide the denominator by 4: \(\frac{12}{4} = 3\)

This means that \(\frac{8}{12}\) simplifies to \(\frac{2}{3}\). Simplifying fractions makes it easier to perform further mathematical operations, especially in more complex algebraic expressions.

Properties of Multiplication

Understanding the properties of multiplication is crucial when tackling algebraic expressions. For fractions, multiplying two fractions involves multiplying their numerators and denominators separately. This property can be expressed with the formula: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\). For example, consider the fractions \(\frac{2}{3}\) and \(\frac{4}{5}\):

• Multiply the numerators: 2 × 4 = 8
• Multiply the denominators: 3 × 5 = 15

Therefore, \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\). Another important aspect is that the order in which you multiply doesn’t matter (commutative property of multiplication). Thus, \(\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}\).

Canceling Common Factors

Canceling common factors is a powerful technique when dealing with multiplication and division of fractions. It involves simplifying an expression before performing further calculations, which can save time and reduce errors. In our original exercise: \(\frac{3 x^{4}}{5 z^{8}} \times \frac{5 z^{8}}{3 x^{4}}\), note that we have common factors in both the numerators and denominators. Here's how to approach it:

• Identify the common factors: Here, they are 3, 5, \(x^4\), and \(z^8\)
• Cancel the common factors: This simplifies both the numerator and the denominator simultaneously

In the expression \(\frac{3 x^{4}}{5 z^{8}} \times \frac{5 z^{8}}{3 x^{4}}\), canceling out the common factors directly gives us: \(\frac{15 x^{4} z^{8}}{15 z^{8} x^{4}} = 1\). This simple approach leads us efficiently to our result.