Problem 3 - Algebra and Trigonometry

Question

When we multiply two powers with the same base, we ______ the exponents. So \(3^{4} \cdot 3^{5}=\) _____.

Step-by-Step Solution

Answer

We add the exponents. So, \(3^4 \cdot 3^5 = 3^9\).

Step 1: Identify the Bases

In the expression \(3^4 \cdot 3^5\), observe that both terms have the same base, which is 3.

Step 2: Apply the Power Rule for Multiplication

When multiplying powers with the same base, we add the exponents. This is known as the Power Rule. For \(3^4 \cdot 3^5\), add the exponents 4 and 5.

Step 3: Calculate the New Exponent

Add the exponents: \(4 + 5 = 9\).

Step 4: Write the Final Expression

Using the Power Rule, combine the bases with the new exponent: \(3^9\).

Key Concepts

Power Rule

The power rule is a fundamental concept in understanding exponents. When dealing with powers, particularly when multiplying them, this rule can help simplify expressions significantly.

The power rule states that if you multiply two powers with the same base, you should simply add the exponents. This is because multiplying the numbers means the base number is being multiplied by itself repeatedly.

Imagine you have the expression \( a^m \cdot a^n \). Here’s what you do step-by-step:

• Identify the base, which is \(a\) in this case.
• Add the exponents \(m\) and \(n\).
  Thus, the expression becomes \( a^{m+n} \).

The power rule helps keep calculations simple and straightforward, helping to solve problems efficiently.

Multiplying Powers with the Same Base

When multiplying powers with the same base, it's important to realize that you are essentially dealing with an extended version of multiplication involving the base number. The concept of 'same base' means that the base number remains constant in both expressions you are multiplying.

Consider the expression \(3^4 \cdot 3^5\) as an example:

• Both numbers share the base 3.
• You then apply the power rule by adding the exponents, which are 4 and 5.

By adding the exponents, you end up with a new single power: \(3^{4+5} = 3^9\). Instead of doing long multiplication of powers, this shortcut makes the process clear and fast.

Understanding this principle is essential for efficiently tackling algebraic problems involving exponents.

Exponent Addition

Exponent addition is a technique that's widely used when applying the power rule. This method stems from the rule where, upon multiplying powers with the same base, the exponents are summed up.

Think of the exponents as the number of times the base is used in multiplication. For instance:

• In \(3^4 \cdot 3^5\), you're effectively multiplying the base, 3, a total of 9 times.

By adding the exponents, 4 and 5, we signify that the base 3 is being multiplied by itself 9 times.

This simplification through adding exponents means less manual multiplication and converting larger expressions into more manageable ones. Furthermore, it highlights the beauty of patterns in mathematics, making it easier for students to see the logic behind algebraic expressions.