Problem 8 - Elementary Algebra

Question

For exercises 7-32, simplify. $$ \left(\frac{8}{5 w+10}\right)\left(\frac{5}{24}\right) $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{1}{3(w + 2)}\).

Step 1 - Factor the denominator

First, factor the denominator in the expression \(\frac{8}{5w+10}\). Notice that \(5w + 10\) can be factored as \(5(w + 2)\). Thus the expression becomes \(\frac{8}{5(w + 2)}\).

Step 2 - Multiply the fractions

Multiply the numerators and the denominators of the fractions \(\frac{8}{5(w+2)}\) and \(\frac{5}{24}\). Therefore, we get: \(\frac{8 \times 5}{5(w+2) \times 24}\).

Step 3 - Simplify the expression

Simplify the fraction by canceling the common factors from the numerator and the denominator. The '5' cancels out in both the numerator and denominator, so we are left with: \(\frac{8}{24(w+2)}\). Then simplify the fraction \(\frac{8}{24}\) to get \(\frac{1}{3}\).

Step 4 - Final expression

Combine the results to get the final simplified expression: \(\frac{1}{3(w + 2)}\).

Key Concepts

Factoring Denominators

Factoring denominators is a crucial step in simplifying algebraic fractions. In our exercise, we had the denominator of the first fraction as \( 5w + 10 \). To simplify this, we can look for common factors in the terms. Here, both 5 and 10 share a common factor of 5.
By factoring out the 5, we get: \[ 5w + 10 = 5(w + 2) \]
This step is essential because it helps to reveal any common factors with the numerator, which will make the simplification process more streamlined later on.

Multiplying Fractions

Once the fractions are prepared, we multiply them. Multiplying fractions involve multiplying the numerators together and the denominators together. For our example:

• Numerators: \ 8 \ and \ 5 \ multiply to give \ 8 \times 5 = 40 \
• Denominators: \ 5(w + 2) \ and \ 24\ multiply to give \ 5(w + 2) \times 24 \

So the fraction becomes: \[ \frac{40}{5(w + 2) \times 24} \] After simplifying common factors, this fraction becomes much easier to handle.

Simplifying Expressions

Simplifying expressions is all about reducing the fraction to its simplest form. In our exercise, after multiplying the fractions, we found: \[ \frac{40}{5(w + 2) \times 24} \] We need to look for common factors in the numerator and the denominator. Here, we can see the '5' in the denominator can be canceled out:
\[ \frac{40}{5 \times 24 \times (w+2)} = \frac{8}{24(w+2)} \]
Next, simplify \ \frac{8}{24}\ to \ \frac{1}{3} \: \[ \frac{8}{24} \rightarrow \frac{1}{3} \] This leaves us with: \[ \frac{1}{3(w + 2)} \]

Canceling Common Factors

Canceling common factors is vital for simplifying fractions. In our exercise, the common factor was '5' between the numerator and the denominator:
\[ \frac{40}{5 \times 24 \times (w+2)} = \frac{8}{24(w+2)} \]
By canceling '5' from both the numerator and denominator, we directly simplify the fraction. This makes the final multiplication and division easier. Remember, to cancel factors, they must be exactly the same in both the numerator and the denominator.
This approach helps you get to the final result quickly and correctly: \[ \frac{1}{3(w + 2)} \]