Problem 14 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{14 x^{3}(7-3 x)}{21 x(3 x-7)}$$

Step-by-Step Solution

Answer

The simplified result of the expression is \(\frac{2x^2}{3}\). Please note that this is assuming that \(x \neq \frac{7}{3}\), to prevent division by zero errors.

Step 1: Apply the Distributive Law

Apply the Distributive Law in the numerator: \(14 x^{3} \times (7 - 3x) = 98 x^{3} - 42 x^{4}\) . The new expression is \(\frac{98 x^{3} - 42 x^{4}}{21 x(3 x - 7)}\).

Step 2: Factor out common elements in the numerator

The common factor in the numerator is \(14x^3\). Factor out to get \(14x^3(7 - 3x) = 14x^3 \times 7 - 14x^3 \times 3x\). Simplifying gives \(14x^3 \times 7\). The expression becomes \(\frac{14x^3(7 - 3x)}{21 x(3 x - 7)}\).

Step 3: Factor out common elements in the denominator

Factor out the common element in the denominator to get \(21 x \times (7 - 3x)\). The expression now becomes \(\frac{14x^3(7 - 3x)}{21 x(7 - 3x)}\).

Step 4: Cancel common factors

Notice that both the numerator and the denominator have the common factor \(7 - 3x\). We can cancel out this common factor, leaving us with the simplified expression \(\frac{14x^3}{21x}\).

Step 5: Simplify the Expression

Further cancelling, we can divide 14 and 21 by their GCF which is 7. That gives us 2 over 3. The x's also cancel to leave \(x^2\). Hence the final simplified expression is \(\frac{2x^2}{3}\).

Key Concepts

Simplification of Fractions

Simplifying fractions means reducing them to their simplest form. Think of fractions like cupcakes. You can remove unnecessary parts until you have the bare essentials. In algebraic fractions, this involves making the expression as simple as possible.

To simplify a fraction, follow these steps:

• Identify and cancel out any common factors in the numerator and denominator.
• Perform any mathematical operations needed to reduce the expression.
• Ensure that no further simplification is possible.

In our example, simplifying the fraction \(\frac{14x^3(7-3x)}{21x(3x-7)}\) involves leveraging the distributive law, factoring, and canceling common terms.

Distributive Law

The Distributive Law is like spreading peanut butter evenly over slices of bread. In math, it's about distributing a single term across terms inside parentheses. This law states that \(a(b+c) = ab + ac\).

In the original problem, we apply the distributive law by multiplying \(14 x^3\) by each term in the parentheses:

• \(14 x^3 \times 7 = 98 x^3\)
• \(14 x^3 \times (-3x) = -42 x^4\)

This gives us the expression \(98 x^3 - 42 x^4\) in the numerator. It's a crucial initial step to simplify and set up the fraction for further reduction.

Factoring

Factoring is like breaking a complicated puzzle into simpler pieces. It's about identifying factors that are common and can be pulled out of an expression.

In the numerator \(98 x^3 - 42 x^4\), we notice that \(14x^3\) is a common factor. By factoring out \(14x^3\), we rewrite the expression as \(14x^3(7 - 3x)\). This makes it easier to handle because we have isolated what we can potentially cancel in the future steps.

Understand that factoring helps us see hidden simplifications and is a vital step in reducing the expression fully.

Canceling Common Factors

Canceling common factors is like tidying up and discarding duplicate items. When we have the same factor in the numerator and the denominator, we can cancel them out.

In our expression, once factored, both the numerator and the denominator have \(7 - 3x\) as a common factor. Removing this from both parts, we simplify the fraction to \(\frac{14x^3}{21x}\).

Further simplification involves dividing 14 and 21 by their greatest common factor (GCF), which is 7, resulting in \(\frac{2x^3}{3x}\). Lastly, canceling \(x\) gives us \(\frac{2x^2}{3}\).

• Canceling is a powerful process that makes equations simpler and more manageable.

Always ensure you've factored completely before canceling to avoid mistakes!