Problem 2 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of the expression is 32.

Step 1: Understand the expression

The expression given is \( 2^3 \cdot 2^2 \). This expression involves multiplying two powers with the same base.

Step 2: Apply the product of powers rule

When multiplying two expressions that have the same base, you can add the exponents. The rule is expressed as \( a^m \cdot a^n = a^{m+n} \). Here, \( a = 2 \), \( m = 3 \), and \( n = 2 \). Thus, \( 2^3 \cdot 2^2 = 2^{3+2} \).

Step 3: Calculate the new exponent

Add the exponents \( 3 \) and \( 2 \) together to get \( 3 + 2 = 5 \). The expression simplifies to \( 2^5 \).

Step 4: Evaluate the expression

Now, calculate \( 2^5 \). This means multiplying 2 by itself 5 times: \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32 \).

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a key concept in the world of exponents that helps simplify expressions when you are dealing with powers of the same base. This rule states that when you multiply two exponents with the same base, you simply add the exponents together. This can be remembered with the formula \( a^m \cdot a^n = a^{m+n} \).

Here’s a simple example: If you have \( 2^3 \cdot 2^2 \), both terms have the base 2. To simplify, you apply the rule by adding the exponents: 3 and 2, resulting in \(2^{3+2} = 2^5\).

This rule greatly simplifies calculations and helps avoid the tedious task of multiplying out each power individually. It is important because it's a fundamental concept that will be used repeatedly in more complicated algebraic expressions.

Multiplication of Powers with Same Base

Multiplying powers with the same base is straightforward once you understand that the base remains the same while you work with the exponents. This principle is rooted in the Product of Powers Rule.

Consider the expression \( 2^3 \cdot 2^2 \). Here, the base 2 does not change; rather, you focus on the exponents: 3 and 2. Following the rule, you simply add these exponents to calculate the product, resulting in \( 2^{3+2} = 2^5 \).

Key steps include:

• Identifying the common base (in this case, it's 2).
• Adding the exponents together.
• Writing the new expression with the same base and the new exponent.

This method not only simplifies expressions but also allows you to handle much larger calculations effectively without tedious multiplication.

Simplifying Exponential Expressions

Simplifying exponential expressions is a crucial skill especially when dealing with complex algebraic expressions. After applying the Product of Powers Rule, as with \( 2^3 \cdot 2^2 \), you take the newly formed power, \( 2^5 \), and evaluate it.

To find the value of \(2^5\), you need to multiply 2 by itself five times: \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32 \).

Simplifying doesn't just mean calculating the final number, but understanding the process. By following these steps:

• Use rules like the Product of Powers to add exponents.
• Express the expression with a single base and an updated exponent.
• Calculate the final expression for its numeric value if required.

Simplifying expressions allows for a faster, more efficient way to work through algebra and provides a solid foundation for tackling even more advanced mathematical problems later on.