Problem 8 - Algebra 2

Question

Simplify each expression. \(\frac{2 a^{2}}{5 b^{2} c} \cdot \frac{3 b c^{3}}{8 a^{2}}\)

Step-by-Step Solution

Answer

\( \frac{3c^{2}}{20b} \)

Step 1: Write the expression

Start with the given expression: \( \frac{2 a^{2}}{5 b^{2} c} \cdot \frac{3 b c^{3}}{8 a^{2}} \). We will simplify by multiplying the numerators together and the denominators together.

Step 2: Multiply the numerators

Multiply the numerators of the fractions: \( 2a^{2} \times 3bc^{3} = 6a^{2}bc^{3} \).

Step 3: Multiply the denominators

Multiply the denominators of the fractions: \( 5b^{2}c \times 8a^{2} = 40b^{2}ca^{2} \).

Step 4: Form the new fraction

Combine the products of the numerators and the denominators to form a new fraction: \( \frac{6a^{2}bc^{3}}{40b^{2}ca^{2}} \).

Step 5: Simplify the fraction

Simplify the fraction by canceling out the common factors in the numerator and the denominator. - Cancel \(a^{2}\) from both numerator and denominator.- Cancel \(b\) from the numerator and one \(b\) from \(b^{2}\) in the denominator.- Cancel \(c\) from the denominator and one \(c\) from \(c^{3}\) in the numerator.- The expression becomes: \( \frac{6c^{2}}{40b} \).

Step 6: Reduce the fraction

Reduce \( \frac{6c^{2}}{40b} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{6c^{2}}{40b} = \frac{3c^{2}}{20b} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its smallest form while retaining the same value. Think of it as making a big fraction smaller in a way that's easier to understand and work with.
In algebra, simplifying fractions involves canceling out common terms in both the numerator and the denominator. These common terms are essentially the same factors present on both sides of the fraction bar.

For example, in the fraction \( \frac{6c^2}{40b} \), the greatest common divisor is 2. This means we can divide both the top (numerator) and the bottom (denominator) by 2.

• Numerator: \( 6c^2 \div 2 = 3c^2 \)
• Denominator: \( 40b \div 2 = 20b \)

The result is the simplified fraction \( \frac{3c^2}{20b} \).
Simplifying is a crucial step because it makes calculations more manageable and results clearer.

Multiplying Fractions

To multiply fractions, you need to multiply the numerators (top numbers) with each other and the denominators (bottom numbers) with each other separately. This method works the same for both regular fractions and algebraic expressions.
It's important to follow this method carefully to ensure accuracy.

Consider the fractions \( \frac{2a^2}{5b^2c} \) and \( \frac{3bc^3}{8a^2} \). To multiply these:

• Multiply the numerators: \( 2a^2 \times 3bc^3 = 6a^2bc^3 \)
• Multiply the denominators: \( 5b^2c \times 8a^2 = 40b^2ca^2 \)

The new fraction formed is \( \frac{6a^2bc^3}{40b^2ca^2} \).
Remember, after multiplying, always check if the fraction can be simplified by factoring out any common terms.

Factoring Common Terms

Factoring common terms is like finding a similar pattern in both the numerator and the denominator and 'canceling' them out to make the fraction simpler.
This is especially useful in algebra because many terms can be broken down into smaller parts that repeat.

Let's see how this works: After multiplying the fractions to get \( \frac{6a^2bc^3}{40b^2ca^2} \), we find repeated factors in the numerator and the denominator. We should look for common factors like \(a^2\), \(b\), and \(c\):

• Cancel the \(a^2\) by dividing both the numerator and denominator by \(a^2\).
• Cancel one \(b\) from the \(b^2\) in the denominator with the \(b\) in the numerator.
• Cancel one \(c\) from \(c^3\) in the numerator with the \(c\) in the denominator.

What's left is \( \frac{6c^2}{40b} \). Factoring common terms simplifies expressions and makes further calculations easier to handle.