Problem 21 - Beginning and Intermediate Algebra

Question

Simplify. $$2 x^{3} \cdot 3 x^{2}$$

Step-by-Step Solution

Answer

The short answer is: \(6x^5\).

Step 1: Rewrite the expression using the properties of exponents

The expression is: \(2x^3 \cdot 3x^2\). Since we are multiplying 2 expressions with the same base, we can apply the rule of exponents: \(a^m \cdot a^n = a^{m+n}\). Here, the base is x, m=3, and n=2. So, we have: $$ (2x^3) \cdot (3x^2) = 2 \cdot 3 \cdot x^{3+2} $$

Step 2: Calculate the sum of the exponents

In the previous step, we found that the expression is equivalent to \(2 \cdot 3 \cdot x^{3+2}\). Now, we just need to find the sum of the exponents. \(3+2\) is equal to 5, so the expression becomes: $$ 2 \cdot 3 \cdot x^5 $$

Step 3: Multiply the coefficients

The expression is now \(2 \cdot 3 \cdot x^5\). We can multiply the coefficients (2 and 3) to get the final simplified expression. \(2 \cdot 3\) is equal to 6, so the expression becomes: $$ 6x^5 $$ The simplified expression is \(6x^5\).

Key Concepts

Simplifying Expressions

When we simplify expressions, our goal is to make them easier to work with. This involves combining like terms and utilizing mathematical properties effectively. This task is particularly crucial in algebra, where expressions can often contain numerous variables, coefficients, and exponents.
Let's take a close look at the expression \(2x^3 \cdot 3x^2\). At first glance, it might look complex, but by simplifying, we aim to express this in the simplest form possible. Simplification often requires following specific rules and steps systematically, as demonstrated in our example.

• Combine Like Terms: In expressions like the one we are simplifying here, terms with the same base (in this case, \(x\)) can be combined using the properties of exponents.
• Calculate: Compute basic arithmetic involving coefficients and simple operations on the exponents.

By systematically simplifying expressions, we find solutions that are easier to interpret and work with further.

Properties of Exponents

Understanding the properties of exponents is essential when working with expressions involving powers and roots. These properties allow us to manipulate expressions with exponents efficiently and with clarity. Learning these rules is foundational in algebra.
Let's examine some key properties of exponents that are demonstrated in the exercise:

• Product of Powers: When multiplying two expressions with the same base, we add the exponents. This is illustrated by the formula \(a^m \cdot a^n = a^{m+n}\). For example, \(x^3 \cdot x^2\) simplifies to \(x^{3+2} = x^5\).
• Power of a Product: When raising a product to an exponent, each part of the product can be raised to the exponent independently.

Using these properties, calculations become much more manageable and less prone to errors. Knowing which property to apply and when is a skill that enhances your problem-solving toolkit.

Multiplying Monomials

Multiplying monomials involves both multiplying their coefficients and using the properties of exponents to simplify the expression. Let's break it down step by step:
First, ensure that you recognize that a monomial is a single term expression, such as \(2x^3\) or \(3x^2\). When multiplying monomials:

• Multiply the Coefficients: Take the numerical parts of the monomials and multiply them together. In our example, \(2 \cdot 3 = 6\).
• Apply Exponential Rules: Use the properties of exponents to handle the variables. Here, \(x^3 \cdot x^2\) becomes \(x^{3+2} = x^5\).

The final simplified form after multiplying these monomials is \(6x^5\). Understanding these steps and consistently applying them will help in quickly simplifying expressions and solving more complex algebraic problems. Make sure always to handle coefficients separately from the variables.