Problem 13 - College Algebra

Question

Evaluate each exponential expression. $$ 2^{2} \cdot 2^{3} $$

Step-by-Step Solution

Answer

The value of the expression \(2^{2} * 2^{3}\) is 32.

Step 1: Identify the base and the exponents

The base is the number 2 and the exponents are 2 and 3. So the expression is \(2^{2} * 2^{3}\).

Step 2: Apply the 'Product of Powers' property

Using the 'Product of Powers' exponent rule, the expression simplifies to \(2^{2+3}\)

Step 3: Simplify the exponent

Adding the powers together gives \(2^{5}\)

Step 4: Calculate the Value

Now calculate the value, which gives 32.

Key Concepts

Product of Powers

The "Product of Powers" property is a powerful tool when dealing with exponential expressions. When two exponential expressions have the same base, you can multiply them by adding their exponents.

For example, consider the expression \(2^{2} \cdot 2^{3}\). Here, both terms share the same base of 2. By using the Product of Powers property, we add the exponents, 2 and 3, resulting in \(2^{2+3}\).

This simplifies the initial multiplication problem into a single expression with the base 2 raised to the fifth power, \(2^{5}\).

This rule can be extended to any numbers with the same base, making calculations quicker and more efficient. It's important to remember that this property can only be used when the bases are identical.

Base and Exponents

In exponential expressions, understanding the base and exponents is essential. The base is the number being multiplied by itself, while the exponent indicates how many times the base is used as a factor.

In the expression \(2^{3}\), the base is 2, and the exponent is 3. This means you multiply the base, 2, by itself three times: \(2 \times 2 \times 2\).

Correctly identifying the base and the exponent in a given expression is the first step in applying rules like the Product of Powers. Without clearly understanding these components, simplifying or evaluating exponential expressions becomes challenging.

Exponents essentially serve as a shorthand for repeated multiplication, so knowing them well can simplify your work with exponential expressions.

Simplifying Exponents

Simplifying exponents involves reducing the complexity of exponential expressions to reveal a more straightforward form or a single value. The step-by-step simplification process involves using exponent rules effectively.

Begin by applying the Product of Powers if you have expressions with the same base. This reduces multiple terms to a single term with a sum of exponents. For example, \(2^{2} \cdot 2^{3}\) becomes \(2^{5}\).

Once the exponents are simplified, evaluate the power by calculating the base raised to the new exponent. In the case of \(2^{5}\), compute \(2 \times 2 \times 2 \times 2 \times 2\).

The final goal of simplification is to find the most concise and accurate representation of an expression. In this example, \(2^{5}\) equals 32, which is the simplest form or value.