Problem 49 - Algebra 2

Question

Simplify each expression. \(\frac{2 x^{3} y}{z^{5}} \div\left(\frac{4 x y}{z^{3}}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x z}{8 y} \).

Step 1: Understand the problem

We need to simplify the complex fraction \( \frac{2 x^{3} y}{z^{5}} \div \left(\frac{4 x y}{z^{3}}\right)^{2} \). This involves dividing two fractions, and dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 2: Find the reciprocal of the divisor

First, we need the reciprocal of \( \left(\frac{4 x y}{z^{3}}\right)^{2} \). By squaring, we obtain \( \frac{16 x^{2} y^{2}}{z^{6}} \). Thus, the reciprocal is \( \frac{z^{6}}{16 x^{2} y^{2}} \).

Step 3: Rewrite the original problem as a multiplication

Rewrite the division of fractions as multiplication: \( \frac{2 x^{3} y}{z^{5}} \times \frac{z^{6}}{16 x^{2} y^{2}} \).

Step 4: Multiply the numerators and the denominators

Multiply the numerators together: \( 2 x^{3} y \times z^{6} = 2 x^{3} y z^{6} \). Multiply the denominators together: \( z^{5} \times 16 x^{2} y^{2} = 16 x^{2} y^{2} z^{5} \).

Step 5: Simplify the resulting expression

Simplify \( \frac{2 x^{3} y z^{6}}{16 x^{2} y^{2} z^{5}} \) by canceling common factors. For \( x \): \( x^{3} / x^{2} = x \). For \( y \): \( y / y^{2} = \frac{1}{y} \). For \( z \): \( z^{6} / z^{5} = z \). Finally, simplify the coefficients: \( 2 / 16 = \frac{1}{8} \).The result is \( \frac{x z}{8 y} \).

Key Concepts

Simplifying Expressions

The process of simplifying expressions aims to make complex algebraic expressions more manageable by reducing them to their simplest form. In the given problem, we begin with the expression \( \frac{2 x^{3} y}{z^{5}} \div \left(\frac{4 x y}{z^{3}}\right)^{2} \). To simplify, we apply several algebraic rules. This involves understanding how to handle operations such as division and multiplication within fractions, and then methodically reducing the expression.

• Convert division into multiplication using reciprocals.
• Multiply numerators and denominators as separate tasks.
• Reduce each part of the expression by identifying and canceling out common factors.

Once we've completed these steps, the expression is in its simplest form.

Reciprocal of a Fraction

The reciprocal of a fraction is what you multiply the fraction by to get the number 1. When a fraction is given, its reciprocal is found by flipping it upside down. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).We use the concept of reciprocals when dividing fractions. Instead of carrying out the division as it is, we simplify the process by multiplying the first fraction by the reciprocal of the divisor fraction. In the problem, after squaring the fraction \( \left(\frac{4 x y}{z^{3}}\right)^{2} \), we got \( \frac{16 x^{2} y^{2}}{z^{6}} \). Its reciprocal is \( \frac{z^{6}}{16 x^{2} y^{2}} \), and this transforms the operation into a multiplication problem.

Multiplying Fractions

Multiplying fractions is a straightforward process that involves two main steps: multiplying the numerators (top numbers) and multiplying the denominators (bottom numbers).**Steps for Multiplying Fractions:**

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

In the expression \( \frac{2 x^{3} y}{z^{5}} \times \frac{z^{6}}{16 x^{2} y^{2}} \), we multiply:- Numerator: \(2 x^{3} y \times z^{6} = 2 x^{3} y z^{6} \)- Denominator: \(z^{5} \times 16 x^{2} y^{2} = 16 x^{2} y^{2} z^{5} \)This gives us a new fraction, ready for further simplification by canceling out common factors.

Canceling Common Factors

Canceling common factors is a critical step in simplifying fractions. After multiplying fractions, you often end up with common factors in the numerator and the denominator, which can be reduced to simplify the fraction.To cancel out common factors, follow these steps:

• Look for variables and numbers that appear in both the numerator and denominator.
• For variables, adjust the exponents by subtracting (e.g., \(x^3/x^2 = x\)).
• For coefficients, divide both by their greatest common divisor (e.g., \(2/16 = 1/8\)).

In the equation \( \frac{2 x^{3} y z^{6}}{16 x^{2} y^{2} z^{5}} \), the common factors were reduced as follows:- \(x^3/x^2 = x\)- \(y/y^2 = \frac{1}{y}\)- \(z^6/z^5 = z\)- \(2/16 = \frac{1}{8}\)Thus, the simplified expression becomes \( \frac{xz}{8y} \). This approach of canceling common factors is essential to ensuring the results are simple and easy to interpret.