Problem 8 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ 2^{3} \cdot 2^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 32.

Step 1: Identify the Base and Exponents

The exercise involves multiplication of powers with the same base. Here, the base is 2, and the exponents are 3 and 2, respectively.

Step 2: Apply the Power of a Power Rule

When multiplying powers with the same base, you add the exponents. Therefore, we calculate: \(2^{3} \cdot 2^{2} = 2^{3+2}\).

Step 3: Simplify the Expression

Perform the addition in the exponent: \(3 + 2 = 5\). This gives us \(2^5\).

Step 4: Calculate the Power

Calculate \(2^5\) by multiplying the base 2 by itself 5 times: \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\).

Key Concepts

Multiplying Powers

Understanding how to multiply powers is essential when working with exponents. In this exercise, you encounter expressions where the base is the same, in this case, the number 2. When multiplying powers that share the same base, a simple rule applies: you keep the base and add the exponents together. This method helps in breaking down more complex problems into simpler calculations.

• For example, in the expression \(2^3 \cdot 2^2\), the base is 2 for both terms.
• To multiply these together, you only need to add the exponents, resulting in \(2^{3+2}\).

This rule allows you to quickly simplify seemingly complex multiplications of powers by focusing on the base and the sum of the exponents.

Power of a Power Rule

The power of a power rule is a crucial concept found in the realm of exponents. This rule comes into play when you are dealing with an exponential expression raised to another exponent. The rule states that you multiply the exponents together. This is slightly different from multiplying powers, where the exponents are added rather than multiplied.

• For example, if you have \((2^3)^2\), this means that 2 raised to the 3rd power is raised again to the 2nd power.
• To find the result, multiply the exponents: \(3 \times 2 = 6\).
• The result would be \(2^6\).

Although this rule wasn't applied to the given exercise directly, it's important to understand the distinction between multiplying powers and the power of powers rule, as it broadens your toolkit for tackling various exponential expressions.

Simplifying Expressions

Simplifying expressions is like tidying up a math problem to make it more understandable and easy to manage. By reducing expressions to their simplest form, calculations become more manageable and less prone to error. The process of simplification also reveals the underlying structure of the problem, making it more intuitive to solve.

• In the exercise \(2^3 \cdot 2^2\), simplify by adding the exponents to get \(2^5\).
• Solving \(2^5\) involves multiplying the base 2 by itself five times, resulting in 32.

By practicing these steps, you become more comfortable with expressions. This adds clarity and efficiency to solving such mathematical exercises.