Problem 43 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ \left(-\frac{3 x^{2}}{2}\right)\left(\frac{4 x}{18}\right) $$

Step-by-Step Solution

Answer

-\frac{x^{3}}{3}

Step 1: Identify the problem

In this problem, the task is to simplify the expression \[ \left(-\frac{3 x^{2}}{2}\right)\left(\frac{4 x}{18}\right) \]

Step 2: Simplify Numerator and Denominator separately

First, carry out the multiplication on the tops (numerators) of the fractions separately from the bottoms (denominators). This gives us:\[ -\frac{3*4*x^{2}*x}{2*18} \]

Step 3: Simplify the expression

Simplify the expression by cancelling out common factors from the numerator and the denominator. This will give:\[ -\frac{12*x^{2+1}}{36} \]

Step 4: Apply Exponent Rules

Applying the rule of exponents where \(x^{a} * x^{b} = x^{a+b}\),we get: \[ -\frac{12*x^{3}}{36} \]

Step 5: Simplify the Fraction

Finally by cancelling the common factors in the numerator and the denominator, the expression will simplify to:\[ -\frac{1*x^{3}}{3} = -\frac{x^{3}}{3}\]

Key Concepts

Multiplication of Fractions

Multiplying fractions may seem a bit tricky at first, but it's quite straightforward once you understand the basic steps. When you multiply two fractions, you multiply the numerators together and then the denominators together.
For the expression \( \left(-\frac{3 x^{2}}{2}\right)\left(\frac{4 x}{18}\right) \), we begin by multiplying the numerators, \which are \ \(-3x^{2}\) and \(4x\). This results in \(-3 \times 4 \times x^{2} \times x = -12x^{3}\).Next, multiply the denominators, which are \(2\) and \(18\). This gives us \(2 \times 18 = 36\).
By putting these results together, we form the combined fraction:

• Numerator: -12x^{3}
• Denominator: 36

The result of multiplying these fractions is therefore \(-\frac{12x^{3}}{36}\).

Exponent Rules

Understanding exponent rules is crucial when dealing with algebraic expressions, such as the ones in our exercise. One of the most important rules is that when you multiply terms with the same base, you add their exponents. This is expressed as: \(x^{a} \times x^{b} = x^{a+b}\).
In our example, \(-\frac{3x^{2} \times 4x}{2 \times 18}\), \we have variables with exponents \ \(x^{2}\) and \(x^{1}\). \(x^{1}\) simply means \(x\), since any number or variable raised to the power of one is the number itself. Thus, \when you multiply them, you add their exponents: \

• \(x^{2} \times x = x^{2+1} = x^{3}\)

So, the expression simplifies to \(-\frac{12x^{3}}{36}\). This application of the exponent rule reduces complexity and prepares the expression for the next simplifying step.

Fraction Reduction

Fraction reduction is all about making a fraction as simple as possible by cancelling out common factors in the numerator and denominator.
In our example, after simplifying the product to \(-\frac{12x^{3}}{36}\), \we need to identify the greatest common factor (GCF) of 12 and 36. The GCF of 12 and 36 is 12.
To simplify, divide both the numerator and the denominator by their GCF:

• Numerator: \(-12x^{3} \div 12 = -1x^{3} = -x^{3}\)
• Denominator: \(36 \div 12 = 3\)

Thus, the fraction reduces to \(-\frac{x^{3}}{3}\).This makes the expression easier to work with and is a crucial step when simplifying algebraic fractions.