Problem 13 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$\frac{6 x}{14} \cdot \frac{2 x^{3}}{5 x^{5}}$$

Step-by-Step Solution

Answer

The simplified form of the given product is \(\frac{6}{35x}\)

Step 1: Simplify the Fractions

First, reduce each of the fractions to its simplest form if possible. Both fractions have common terms in the numerator and the denominator which can be divided out: \(\frac{6x}{14} = \frac{3x}{7}\) (dividing by 2) and \(\frac{2x^{3}}{5x^{5}} = \frac{2}{5x^{2}}\) (reducing exponents).

Step 2: Multiplication of Fractions

Next, multiply the numerators together and then the denominators together to form a new fraction: \(\frac{3x}{7} \cdot \frac{2}{5x^{2}} = \frac{3x \cdot 2}{7 \cdot 5x^{2}}\).

Step 3: Simplify the Resulting Fraction

Carry out multiplication in the numerator and the denominator: \(\frac{6x}{35x^{2}}\). Notice that x can still be divided out from numerator and the denominator, resulting in \(\frac{6}{35x}\).

Key Concepts

Multiplication of Fractions

To multiply fractions, follow these simple steps: multiply the numerators to find the new numerator, and the denominators to find the new denominator. Let's break it down:

• Numerators: Take the top numbers from each fraction and multiply them. For example, in \( \frac{3x}{7} \) and \( \frac{2}{5x^2} \), the numerators are \( 3x \) and \( 2 \), and their product is \( 3x \times 2 = 6x \).
• Denominators: Multiply the bottom numbers in the same way. So, \( 7 \times 5x^2 = 35x^2 \).

Now you have a new fraction: \( \frac{6x}{35x^2} \).
This technique is straightforward and works for all kinds of fractions, even those with variables. Remember to multiply across the numerators and denominators separately to avoid errors.

Exponent Rules

Exponents can seem tricky, but they're just repeated multiplications. Here, we'll see how they help simplify expressions.
When multiplying variables with exponents, if the bases are the same, add the exponents. For example:

• \( x^3 \times x^2 \) means you have three "x" multiplied by two more, giving \( x^{3+2} = x^5 \).

In division, subtract exponents of like bases. This is handy when reducing fractions:

• If you have \( \frac{x^3}{x^5} \), \( x^5 \) can be simplified by subtracting the exponents: \( x^{3-5} = x^{-2} \), or expressed positively as \( \frac{1}{x^2} \).

By understanding these rules, you can manage exponents and make algebraic expressions easier to handle.

Reducing Fractions

Reducing fractions allows you to simplify complex expressions. It involves eliminating common factors from the numerator and the denominator.
First, look for common factors. These are numbers or terms that both the numerator and the denominator share.

• For example, in \( \frac{6x}{14} \), both 6 and 14 can be divided by 2, resulting in \( \frac{3x}{7} \).

In fractions with variables, apply the same principle:

• Take \( \frac{2x^3}{5x^5} \). By dividing both the numerator and the denominator by their common term \( x^3 \), you get \( \frac{2}{5x^2} \).

This step often makes further calculations easier and helps to find the simplest form, as in our final result, \( \frac{6}{35x} \). Keeping fractions simple can prevent errors and make your math work easier to follow.