Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \frac{\left(2 x^{3}\right)\left(3 x^{2}\right)}{\left(x^{2}\right)^{3}} $$

Step-by-Step Solution

Answer

\( \frac{6}{x} \)

Step 1: Multiply the Numerator

First, multiply the terms in the numerator: \((2x^3) \times (3x^2) = 2 \times 3 \times x^{3+2} = 6x^5. \)

Step 2: Simplify the Denominator

Next, simplify the denominator by applying the power rule: \((x^2)^3 = x^{2 \times 3} = x^6. \)

Step 3: Simplify the Fraction

Now divide the simplified numerator by the simplified denominator:\(\frac{6x^5}{x^6} = 6x^{5-6} = 6x^{-1}. \)

Step 4: Final Simplification

Express the final answer with positive exponents:\( 6x^{-1} = \frac{6}{x}. \)

Key Concepts

Numerator Multiplication

In algebra, when we deal with fractions whose numerators need to be multiplied, the problem can appear more daunting than it really is. Multiplying the terms in the numerator is often the first step. Here's how you do it.
In our given exercise, we're asked to multiply two expressions: \((2x^3) \times (3x^2)\).
To multiply, follow these simple steps:

• First, multiply the coefficients (the numbers in front of variables). In this case, multiply 2 by 3. The result is 6.
• Next, focus on the variables. Since both terms have an "x," you add their exponents when multiplying same bases, giving you \(3+2=5\).

Thus, the result of multiplying these terms is \(6x^5\). Breaking it down into parts (coefficients and variables) makes the multiplication process clear and systematic.

Power Rule in Algebra

The power rule is essential when dealing with exponents, especially when simplifying expressions raised to a power. It tells us that when an exponent is raised to another power, you multiply the exponents.
For example, in the expression \((x^2)^3\), you're taking \(x^2\) and multiplying the exponent 2 by 3.

• Begin by identifying the base, which here is "\(x\)."
• Then, multiply the exponents: \(2 \times 3 = 6\).

After applying the power rule, you simplify \((x^2)^3\) to \(x^6\). The power rule is a powerful tool that simplifies expressions in just a step or two.

Exponent Subtraction

Exponent subtraction occurs typically when you simplify fractions in algebra. It's applied when you divide the same bases with exponents by subtracting the exponents.Consider our fraction \(\frac{6x^5}{x^6}\). To simplify this:

• Identify that "x" is the common base.
• Subtract the exponents in the numerator and the denominator. So here, \(5 - 6 = -1\).

Therefore, \(\frac{6x^5}{x^6} = 6x^{-1}\). This method simplifies expressions effectively by focusing on shared bases and manipulating their exponents through subtraction.

Positive Exponent Conversion

In algebraic simplification, we prefer expressing answers with positive exponents as they are easier to interpret. Let's convert negative exponents into positive ones. Take our resulting expression from the previous step, \(6x^{-1}\).To convert this, remember: A negative exponent implies the reciprocal of the base to a positive exponent. So, \(x^{-1}\) can be rewritten as \(\frac{1}{x}\). Therefore, our expression becomes:

• Rewrite \(6x^{-1}\) as \(6 \times \frac{1}{x}\).
• Simplify to \(\frac{6}{x}\).

Thus, the expression \(6x^{-1}\) converts to \(\frac{6}{x}\), which is often more intuitive and aligned with conventional mathematical presentation.