Question

Simplify the expression. $$ 3^2 \cdot 3^4 $$

Short answer

\[3^6\] is the simplified version of the expression.

Step-by-step solution

Identify the bases and exponents

The bases are the same (3) and the exponents are 2 and 4.

Apply the law of exponentials

When the bases are the same and the operation is multiplication, we add the exponents. The new expression becomes \(3^{2+4}\).

Simplify the new expression

The final expression simplifies to \(3^6\) after adding the exponents together.

Key concepts

Law of Exponents

Understanding the law of exponents, also known as exponent rules, is essential for simplifying exponential expressions efficiently. These laws give us specific rules for performing operations with powers.

The law we applied in the exercise is one of the most basic: when multiplying two exponents with the same base, you keep the base and add the exponents together. Mathematically, this law is expressed as:\[\begin{equation}a^m \times a^n = a^{m+n}ewline ewline ewline ewline \end{equation}\]This rule makes it fast and easy to combine expressions like the one we encountered, where we had the same base (3) and simply needed to add the exponents (2 and 4) to get our simplified expression, which was \(3^{2+4}\) or \(3^6\).

To get a solid grip on this, practice with various bases and exponents. Remember, the base must be the same for this specific law to apply.

Base and Exponent

The base and exponent are two parts of an exponential expression. The base is the number that is being multiplied by itself, and the exponent tells us how many times the base is used as a factor in the multiplication.

For example, in the expression \(3^2\), 3 is the base, and 2 is the exponent, which tells us that 3 is multiplied by itself once because we start counting from zero for the number of multiplications. If we write it out, it looks like this: \(3 \times 3\), which equals 9.

Recognizing the base and exponent is a crucial first step in simplifying expressions, as it helps us determine which laws of exponents we should apply.

Exponential Multiplication

Exponential multiplication deals with situations where we multiply expressions with exponents. As illustrated in the exercise, when the bases being multiplied are the same, the process is straightforward: add the exponents.

In general terms, if we have \(a^m\) and \(a^n\), the multiplication result is \(a^{m+n}\). An effective way to get comfortable with this process is to practice multiplying different bases and exponents, always ensuring that the bases match when applying this rule.

Once you grasp this concept, you'll see patterns such as \(2^3 \times 2^4 = 2^{3+4} = 2^7\), which simplifies to 128. It's a powerful tool that helps both in simple and complex algebraic operations.