Problem 5 - Beginning and Intermediate Algebra

Question

Multiply. $$\frac{22}{15 r^{2}} \cdot \frac{5 r^{4}}{2}$$

Step-by-Step Solution

Answer

The short answer is: \(\frac{11r^2}{3}\).

Step 1: Multiply the numerators

First, we have to multiply the numerators of both fractions: \(\frac{22}{15r^2}\) and \(\frac{5r^4}{2}\). \(22 \times 5r^4 = 110r^4\)

Step 2: Multiply the denominators

Now, let's multiply the denominators of the given fractions: \(15r^2 \times 2 = 30r^2\)

Step 3: Write the product of the fractions

Now that we have the product of the numerators and denominators, we can write the new fraction: \(\frac{110r^4}{30r^2}\)

Step 4: Simplify the fraction

Let's simplify the fraction by cancelling any common factors. We can see that both the numerator and denominator have a common factor of \(10\), and \(r^2\). \(10\) divides \(110\) - \(11\) times and \(10\) divides \(30\) - \(3\) times. Additionally, we can divide both the numerator and denominator by \(r^2\). So, our simplified fraction is: \(\frac{11r^{(4-2)}}{3}\)

Step 5: Write the final answer

Now we have the simplified fraction: \(\frac{11r^2}{3}\) So, the answer is \(\frac{11r^2}{3}\).

Key Concepts

Multiplying Fractions

Multiplying fractions might seem a bit daunting at first, especially with algebraic expressions involved, but it follows straightforward rules that can be mastered step-by-step. When we multiply fractions, the process involves two main actions:

• Multiply the numerators (the top numbers) of the fractions together.
• Multiply the denominators (the bottom numbers) together.

For example, in the original problem, the fractions \(\frac{22}{15r^2}\) and \(\frac{5r^4}{2}\) require multiplying both their numerators and denominators:

• Numerator: \(22 \times 5r^4 = 110r^4\)
• Denominator: \(15r^2 \times 2 = 30r^2\)

The result of multiplying these fractions, before simplification, is \(\frac{110r^4}{30r^2}\). Remember, always multiply straight across - numerator with numerator, and denominator with denominator.

Simplifying Fractions

After multiplying fractions, simplifying the result helps make it more comprehensible and tidy. Simplifying involves canceling out the greatest common factors from both the numerator and the denominator, which reduces the fraction to its simplest form. In the context of algebraic fractions, this may also involve reducing powers of variables.
In our example of \(\frac{110r^4}{30r^2}\), here’s how we simplify:

• First, look for numerical common factors: \(10\) divides both \(110\) and \(30\). Dividing them gives us \(\frac{11r^4}{3r^2}\).
• Then, simplify the variables: Since \(r^4\) and \(r^2\) share the common base \(r\), divide out \(r^2\) from both. This leaves \(\frac{11r^{4-2}}{3}\), or \(\frac{11r^2}{3}\).

Now, we have our fraction in its simplest form! Simplifying not only gives a cleaner answer but also helps understand the relationship between parts of the expression more clearly.

Algebraic Expressions

Understanding algebraic expressions is crucial when working with fractions that include variables. An algebraic expression contains numbers, variables (like \(r\)), and operations (like multiplication or addition). These variables represent unknown or varying quantities.
When dealing with algebraic fractions, such as \(\frac{22}{15r^2}\), it’s important to treat variables with care:

• Remember that variables can be factored just like numbers.
• Use the laws of exponents: For instance, when multiplying \(r^4\) by \(r^2\), add their exponents to get \(r^{4+2} = r^6\).
• Simplification often involves reducing the powers of these variables.

Handling algebraic expressions becomes simpler when you understand these foundational principles. They allow you to manipulate and solve fractions that initially seem complex.