Problem 14 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$3^{3} \cdot 3^{2}$$

Step-by-Step Solution

Answer

The result of the expression \(3^{3} \cdot 3^{2}\) is 243.

Step 1: Understanding the problem

We are given an exponential expression \(3^{3} \cdot 3^{2}\). This operation involves the same base (3) raised to different powers, 3 and 2.

Step 2: Applying the law of exponents

The law of exponents states that when multiplying like bases, the exponents should be added. This can be expressed as \(a^{m} \cdot a^{n} = a^{m+n}\). Therefore, our expression changes to \(3^{3+2}\).

Step 3: Calculating the result

We now perform the addition of the exponents, leading to \(3^{5}\). This means we have 3 multiplied by itself 5 times. Therefore, the result of our expression is \(3^{5} = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243\).

Key Concepts

Exponential Expression

An exponential expression is a mathematical notation that involves numbers known as bases raised to a certain power called exponents. In this expression, the base is the number that is multiplied by itself, while the exponent indicates how many times the base is used as a factor.
In our original example, the exponential expression is given as \(3^3 \cdot 3^2\). Here, 3 is the base and it is raised to the powers of 3 and 2.
This means the base (3) will be multiplied itself according to the respective exponents:

• \(3^3\) means \(3 \cdot 3 \cdot 3\).
• \(3^2\) means \(3 \cdot 3\).

Such expressions allow us to simplify the representation of repeated multiplication of the same number, making it easier to handle in calculations and algebraic expressions.

Laws of Exponents

The laws of exponents are essential rules in algebra that help us simplify expressions and solve equations involving powers. These rules dictate how we handle operations of exponentiation efficiently. Let's focus on one of these laws that applies to our problem.
The multiplication of exponents law states:

• When you multiply two exponential expressions with the same base, you keep the base and add the exponents: \(a^m \cdot a^n = a^{m+n}\).

This simplifies calculational work by reducing a potentially complex problem into manageable steps.
To exemplify this, in our original problem, we multiply two exponents with the base 3. Applying the law, we add the exponents (3 and 2) instead of performing each multiplication separately, simplifying the expression to \(3^{3+2}\).

Multiplication of Exponents

Multiplication of exponents occurs when dealing with expressions sharing a common base. Knowing how to correctly combine these is crucial for simplifying expressions and calculating results swiftly.
For an expression like \(3^3 \cdot 3^2\):

• Recognize that both numbers have the same base, 3.
• Instead of multiplying \(3 \cdot 3 \cdot 3\) and \(3 \cdot 3\) separately, we apply the law of exponents to keep computations simple.

We combine the exponents to get \(3^{3+2} = 3^5\), meaning 3 is multiplied by itself five times.
This allows for a quick calculation yielding \(3^5 = 243\). Multiplying exponents in this way not only reduces errors but also saves time during calculations.