Problem 40 - Precalculus: Functions and Graphs

Question

Simplify $$\left(3 x^{1 / 2}\right)\left(-2 x^{3 / 2}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-6x^2\).

Step 1: Understand the Problem

We have the expression \( (3x^{1/2})(-2x^{3/2}) \). We need to simplify it by multiplying the coefficients and adding the exponents of the same base.

Step 2: Multiply the Coefficients

Identify and multiply the coefficients: \(3\) and \(-2\). This gives us \( 3 \times (-2) = -6 \).

Step 3: Add the Exponents

For the variables with the same base \(x\), add their exponents: \((1/2) + (3/2) = 4/2 = 2\).

Step 4: Write the Simplified Expression

Combine the results from the previous steps: the simplified expression is \(-6x^2\).

Key Concepts

Multiplying Coefficients

In algebra, when multiplying expressions, particularly those with coefficients, it's important to tackle each part separately. Coefficients are the numerical part of terms in an expression that are multiplied by variables. For our exercise, we have the coefficients 3 and -2. The rule is straightforward: simply multiply these numbers together. This involves a basic arithmetic operation:

• Multiply 3 and -2 to get -6.

Coefficients don't have variables attached to them, so they are easier to handle compared to other parts of the expression. This multiplication step is crucial before moving to involve the exponents. Familiarizing oneself with similar simple arithmetic can help build confidence in handling algebraic expressions.

Adding Exponents

Exponents may seem daunting, but understanding them is key to algebra. In the given expression, we aim to multiply terms with the same base, in this case, the base is \(x\). Each of these terms has an exponent attached.In our exercise, we have \(x^{1/2}\) and \(x^{3/2}\). The rule for multiplying like bases is to add the exponents:

• Add \(1/2\) and \(3/2\) together.

This results in \(4/2\), which simplifies to 2. This means the exponents can be combined into a single term, \(x^2\). Exponent addition works because multiplying powers of the same base extends the degree of that base in the expression. This method of adding exponents is consistent whenever you see like bases in algebra.

Exponents Rules

Mastering the rules of exponents is pivotal to simplifying expressions. There are a few fundamental rules to remember:

• Product of Powers Rule: To multiply powers with the same base, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Power of a Product Rule: To apply an exponent to a product, apply it to each factor: \((ab)^m = a^m \cdot b^m\).

In our context, we applied the Product of Powers rule, ensuring that we added the exponents \((1/2 + 3/2)\) to achieve \(x^2\). Understanding these rules not only helps in algebra but lays the groundwork for more advanced math topics.

Simplified Expressions

The goal of simplifying expressions is to reduce them to their most concise form while retaining the same value or meaning. The original expression \((3x^{1/2})(-2x^{3/2})\) is simplified to \(-6x^2\). This simplification comes from:

• Multiplying the coefficients to get \(-6\).
• Adding the exponents of like bases to get \(x^2\).

This concise expression is not only easier to understand but is also more functional in equations where the variable \(x\) might need to be solved for. Simplified expressions are critical for clear mathematical communication, problem-solving, and even in applied scenarios like calculating areas or modeling real-world problems.