Question

Multiply and simplify. $$(2 a)\left(3 a^{2}\right)$$

Short answer

The simplified expression is 6a^3.

Step-by-step solution

Multiply the Coefficients

Multiply the numerical coefficients together. In this case, multiply 2 by 3.

Multiply the Variables

Multiply the variables by adding the exponents. Since both terms have the variable 'a', you add the exponents. The first term has an exponent of 1 (implied) and the second term has an exponent of 2.

Combine the Results

Combine the results from Step 1 and Step 2 to obtain the simplified expression.

Key concepts

Simplifying Algebraic Expressions

Simplifying an algebraic expression is like decluttering a room; it's about making the expression as neat and as easy to understand as possible. An algebraic expression can look complex with all its variables, coefficients, and exponents. However, the key principle is to combine like terms and use operations such as addition, subtraction, multiplication, and division appropriately.

When simplifying expressions, always combine like terms, which are terms that have the exact same variables raised to the same power. For example, in the expression \(3x^3 + 2x^2 - x^3\), the like terms are \(3x^3\) and \( - x^3\). After combining these like terms, you would be left with \(2x^3 + 2x^2\), a much simpler expression.

Multiplying Coefficients

Think of coefficients as the numbers that give weight to the variables they are attached to. To multiply coefficients, you'll need to follow 'number rules.' This means treating the coefficients as plain old numbers and multiplying them together, clear and simple.

Let's look at our exercise example: \(2a * 3a^2\). Here, the coefficients are 2 and 3. To multiply them, you simply do \(2 * 3 = 6\). Remember, only the numerical parts of terms are considered coefficients, so we leave the variable 'a' out of this multiplication for now. \(2 * 3\) gives us 6, which will serve as our new coefficient for the variables we'll handle next.

Exponent Rules

Exponents can feel like a tricky puzzle initially, but a few key rules will help you to effortlessly navigate through them. An important rule when multiplying variables is the 'Product of Powers rule,' which states that when you multiply two terms with the same base, you can add their exponents.

In the example \(2a * 3a^2\), we have two 'a' terms: \(a^1\) and \(a^2\); the first exponent is implied. Following the rule, we add the exponents: \(1 + 2 = 3\). Therefore, \(a * a^2\) becomes \(a^3\). Now, we can conclude that multiplying \(2a\) and \(3a^2\) yields \(6a^3\), since \(2 * 3 = 6\) (as covered in the previous section about coefficients), and \(a^1 * a^2 = a^{1+2} = a^3\).