Problem 4 - Beginning and Intermediate Algebra

Question

Find each product and simplify if possible. $$ \frac{6 x^{2}}{10 x^{3}} \cdot \frac{5 x}{12} $$

Step-by-Step Solution

Answer

The product is \( \frac{1}{4} \).

Step 1: Multiply the Numerators

First, multiply the numerators of the two fractions: \( 6x^2 \) and \( 5x \). This gives you \( (6x^2) \times (5x) = 30x^3 \).

Step 2: Multiply the Denominators

Now, multiply the denominators of the two fractions: \( 10x^3 \) and \( 12 \). This results in \( (10x^3) \times (12) = 120x^3 \).

Step 3: Formulate the New Fraction

Combine the results from Steps 1 and 2 to form the new fraction: \( \frac{30x^3}{120x^3} \).

Step 4: Cancel Common Terms and Simplify

Identify common terms in the numerator and the denominator. Both \( 30x^3 \) and \( 120x^3 \) can be divided by \( 30x^3 \): \( \frac{30x^3}{120x^3} = \frac{1}{4} \).

Step 5: Final Simplified Form

After canceling common terms, the simplified form of the product is \( \frac{1}{4} \).

Key Concepts

Numerator Multiplication

When dealing with two fractions, before you can find their product, you need to multiply their numerators. The numerators are the top parts of the fractions. In our exercise, the numerators are \(6x^2\) and \(5x\). To perform the multiplication, simply multiply the coefficients and add the exponents of like variables, in this case, \(x\). This follows the rule that

• \(a^m \times a^n = a^{m+n}\)

For our problem, \(6x^2 \times 5x\) becomes \(30x^{2+1} = 30x^3\). Ensuring you multiply correctly here sets the foundation for the correct answer.

Denominator Multiplication

Just like with the numerators, multiply the denominators, which are the bottom parts of the fractions. In this exercise, our denominators are \(10x^3\) and \(12\). Again, multiply the coefficients and sum the exponents of similar terms. This is an important step because the resulting product will be part of the fraction you form later.

• \(10 \times 12 = 120\)
• \(x^3\) remains as \(x^3\) since it is multiplied by \(1\)

So when multiplied, the resulting denominator is \(120x^3\). By correctly handling multiplication of the denominators, you'll accurately form the complete fraction for the next steps.

Simplifying Fractions

Once you have your new fraction, you must simplify it. This means reducing it to its simplest form. Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common factor (GCD). In our example, the fraction \(\frac{30x^3}{120x^3}\) is formed after multiplying the numerators and denominators.Identify the GCD of the coefficients 30 and 120. The largest number that divides both evenly is 30. Also, since both terms have \(x^3\), we can cancel \(x^3\) from both. This sets up our fraction nicely for reducing.

Canceling Common Terms

The final step to simplifying a fraction is canceling common terms. This means you cross out any terms that appear in both the numerator and the denominator. These are essentially factors that can be divided out without changing the fraction's value. For \(\frac{30x^3}{120x^3}\), you divide both numerator and denominator by \(30x^3\). This simplifies the fraction to \(\frac{1}{4}\), by reducing both 30 divided by 30 to 1 and 120 divided by 30 to 4. Canceling common terms is key to simplifying the fraction correctly and ensuring you get the right answer!