Problem 15 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{18 x^{6}}{7} \cdot \frac{1}{4 x^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $\frac{18x^6}{7} \cdot \frac{1}{4x^2}$. Answer: The simplified expression is $\frac{9}{14}x^4$.

Step 1: Multiply the numerators

First, we need to multiply the numerators of the two fractions: $$ (18x^6) \cdot (1) = 18x^6 $$

Step 2: Multiply the denominators

Next, we need to multiply the denominators of the two fractions: $$ (7) \cdot (4x^2) = 28x^2 $$

Step 3: Combine the results

Now, we need to combine the results from Steps 1 and 2 to get the resulting fraction: $$ \frac{18x^6}{28x^2} $$

Step 4: Simplify the fraction

Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor: $$ \frac{18x^6}{28x^2} = \frac{9x^6}{14x^2} $$ Now, we can simplify the exponents: subtract the exponent of the denominator from the exponent of the numerator: $$ \frac{9x^6}{14x^2} = \frac{9}{14}x^{6-2} = \frac{9}{14}x^4 $$ So, the simplified expression is: $$ \frac{9}{14}x^4 $$

Key Concepts

Multiplying Fractions

When it comes to multiplying fractions, the process is straightforward. You take the two fractions you need to multiply and simply multiply the numerators with each other and the denominators with each other. In the given exercise, the fractions are \( \frac{18x^6}{7} \) and \( \frac{1}{4x^2} \). Start by multiplying the numerators: \( 18x^6 \times 1 = 18x^6 \). For the denominators, you multiply \( 7 \) by \( 4x^2 \) to get \( 28x^2 \). This gives you a new fraction: \( \frac{18x^6}{28x^2} \). Two main steps to remember are:

• Multiply the numerators together.
• Multiply the denominators together.

Breaking down the multiplication of fractions into these smaller parts helps make the process easier to manage. Just remember that you're essentially dealing with two teams - the numerators and the denominators - and you'll multiply each team's strengths together.

Dividing Fractions

Dividing fractions is slightly different from multiplying them. The key step involves turning division into multiplication, which we achieve by using the reciprocal. When you divide by a fraction, you actually multiply by its reciprocal. This means you flip the numerator and the denominator of the fraction you are dividing by. For example, if you need to divide by \( \frac{1}{4x^2} \), you multiply by \( \frac{4x^2}{1} \). In our exercise, even though division was not explicitly asked, understanding this can make any division operation with fractions less daunting. The formula to remember is:

• Division of \( a/b \) by \( c/d \) is equivalent to multiplication of \( a/b \) by \( d/c \).
• Convert the division operation to multiplication by using the reciprocal of the divisor.

By turning division into multiplication tasks, you're simplifying what could seem complicated equations into something more approachable.

Simplifying Expressions

Simplifying expressions is an essential part of algebra that can make your final answer neat and easier to interpret. When you reached \( \frac{18x^6}{28x^2} \) from multiplication, you need to simplify this. Here, you would find the greatest common divisor of the coefficients, which is \( 9 \) for \( 18 \) and \( 28 \), simplifying the coefficients to \( \frac{9}{14} \). Simultaneously, simplify the powers of \( x \). Since \( x^6 \) divided by \( x^2 \) is \( x^{6-2} \), you get \( x^4 \). Therefore, the expression simplifies to \( \frac{9}{14}x^4 \). Here’s a quick guide:

• Find common factors between numerators and denominators.
• Reduce coefficients by dividing by their greatest common divisor.
• Simplify variable exponents by subtracting the exponents in the denominator from those in the numerator.

Simplifying expressions not only tidies up your answers but is also a useful skill to check for errors or unnecessary complexity in algebraic problems.