Problem 63 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(5^{2}\right)(8)\left(2^{0}\right) $$

Step-by-Step Solution

Answer

200

Step 1: Simplify each component

First, simplify each component of the expression where possible. We have the expression \( (5^{2})(8)(2^{0}) \). Here, \( 5^{2} = 25 \) and \( 2^{0} = 1 \) because any number to the power of 0 is 1. Thus, the expression becomes \( 25 \times 8 \times 1 \).

Step 2: Calculate the multiplication

Now multiply the simplified numbers together. Start with the first two terms: \( 25 \times 8 \). Calculating this gives \( 200 \). Since multiplying by 1 doesn’t change the outcome, we have \( 200 \times 1 = 200 \).

Step 3: Express the final result with positive exponents

The final result is \( 200 \), which is already in its simplest form with positive exponents only. There are no exponents remaining to adjust.

Key Concepts

Understanding Positive Exponents

Positive exponents indicate how many times a number, known as the base, is multiplied by itself. For example, the expression \( 5^2 \) means that the base \( 5 \) is multiplied by itself 2 times, resulting in \( 5 \times 5 = 25 \). When simplifying expressions, always ensure that exponents are presented as positive values for clarity and consistency.
- Positive exponents show repeated multiplication.- The base raised to a positive exponent implies "multiply the base by itself a number of times equal to the exponent."
In our example, we simplified \( (5^2)(8)(2^0) \) by converting \( 5^2 \) to 25 and recognizing \( 2^0 \) as 1. Both of these are straightforward applications of positive exponents and their rules. Highlighting these conversions is key to mastering expression simplification.

Simplifying Multiplication of Expressions

When simplifying expressions involving multiplication, you must combine numbers through multiplication, one step at a time. This step-by-step approach ensures accuracy and helps manage more complex expressions.

• Identify all components: Break larger expressions into smaller parts that can be simplified separately.
• Use basic arithmetic operations: Start with multiplications and then perform additional operations as needed.
• Keep track of all intermediate steps to prevent any mistakes in calculations.

In the exercise, we started by recognizing that \( 5^2 \) equals 25 and \( 2^0 \) equals 1. We then multiplied these values with 8. We calculated \( 25 \times 8 = 200 \), and since multiplying by 1 does not change the product, it simplifies directly to 200. By breaking down each step, complex expressions become more manageable.

Applying Exponent Rules

Exponent rules are useful for simplifying expressions involving powers. Key rules include:

• Any number raised to the power of 0 is 1 (e.g., \( a^0 = 1 \)). This is shown in our exercise where \( 2^0 = 1 \).
• When the base is raised to a power greater than 0, multiply the base by itself as many times as indicated by the exponent (e.g., \( a^n = a \times a \times \ldots \times a \), \( n \) times).

Understanding and applying these rules can speed up the simplification process by making it easier to convert complex components into simpler ones. In essence, recognizing these rules in our original equation helped us transform \( (5^2)(8)(2^0) \) into its simplest form, 200. By identifying these rules early, expressions can be simplified quickly and accurately.