Problem 15 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Example 1. $$ \frac{3 a}{10} \cdot \frac{2}{15 a^{4}} $$

Step-by-Step Solution

Answer

The simplified product is \( \frac{1}{25a^3} \).

Step 1: Understand the Expression

We have two fractions that we need to multiply: \( \frac{3a}{10} \) and \( \frac{2}{15a^4} \). We can multiply two fractions by multiplying their numerators and denominators separately.

Step 2: Multiply the Numerators

The numerators are \( 3a \) and \( 2 \). Multiply them: \[ 3a \times 2 = 6a \]

Step 3: Multiply the Denominators

The denominators are \( 10 \) and \( 15a^4 \). Multiply them: \[ 10 \times 15a^4 = 150a^4 \]

Step 4: Write the Combined Fraction

Combine the results from steps 2 and 3 into a single fraction:\[ \frac{6a}{150a^4} \]

Step 5: Simplify the Fraction

To simplify, divide the numerator and denominator by their greatest common factor. The GCF of 6 and 150 is 6, and since we have \( a \) in the numerator and \( a^4 \) in the denominator, divide the terms as follows:\[ \frac{6a}{150a^4} = \frac{6 \div 6 \cdot a^{1-1}}{150 \div 6 \cdot a^{4-1}} = \frac{1}{25a^3} \]

Step 6: Final Result

The simplified form of the product of the two fractions \( \frac{3a}{10} \) and \( \frac{2}{15a^4} \) is: \[ \frac{1}{25a^3} \]

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form. A fraction is in its simplest form when the numerator and the denominator are as small as possible, meaning they have no common factors besides 1.

To simplify a fraction, follow these steps:

• Find the Greatest Common Factor (GCF): Determine the largest number that divides both the numerator and the denominator without leaving a remainder. This will help in reducing the fraction effectively.
• Divide Both Terms: Divide both the numerator and the denominator by their GCF. This reduces the fraction to its simplest form.

By following these steps, you are guaranteed to make both parts of the fraction as small as possible, and you ensure the fraction is easier to work with in calculations. It's especially handy in making complex algebraic fractions more manageable.

Greatest Common Factor

The Greatest Common Factor, or GCF, is an important concept for simplifying fractions. It is the largest factor that two numbers share. Finding the GCF is crucial because it helps in reducing fractions by canceling out shared factors between the numerator and the denominator.

Here's how to find the GCF:

• List the Factors: Write down all the factors for each of the numbers you are working with.
• Identify Common Factors: Look for numbers that appear in both factor lists.
• Choose the Largest Factor: The biggest number that appears in both lists is the GCF.

For algebraic fractions, like the example problem, you also need to consider variables. The GCF will be the product of the largest number and the lowest power of any variable common to both the numerator and the denominator.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions but include variables like x or y, or in our case, a. Processing and simplifying these can be more complex due to these additional elements.

Here are some tips for working with algebraic fractions:

• Factor Algebraically: You can often factor both the numerator and the denominator to simplify the fraction by cancelling common factors.
• Use GCF Appropriately: Just as you do with numbers, identify the GCF for the coefficients and the variable parts. This might include dividing the variable powers, as demonstrated by subtracting the exponent in the denominator from the numerator like in the given example.
• Handling Exponents: Remember the rules for exponents. When dividing terms with the same base, subtract the exponents. This is key to simplifying fractions with variables.

Algebraic fractions need more attention, but by systematically applying the rules for factoring and simplifying, they become much easier to manage.