Problem 24 - Algebra 1: Concepts and Skills

Question

Evaluate the power. \(5^{4}\)

Step-by-Step Solution

Answer

Therefore, \(5^{4} = 625\)

Step 1: Understanding the Concept of Exponentiation

When we write \(5^{4}\), it means we're multiplying the number 5 by itself 4 times. Hence, it can be expanded as \(5 * 5 * 5 * 5\). It's a quick way of saying multiply 5 by itself 4 times.

Step 2: Performing the Multiplication

Perform the multiplication operation: Start with the first two 5s. Multiply them, yielding 25. Then multiply 25 by 5, which equals 125. Multiply 125 by the last 5, giving us the final solution of 625.

Key Concepts

Power

The concept of a "power" in mathematics is a way to simplify the expression of repeated multiplication. In our exercise, we see this with the notation \(5^{4}\). Here, the base is 5, and the exponent is 4. This tells us to take the base, 5, and multiply it by itself a total of four times. In other words, calculate \(5 \times 5 \times 5 \times 5\). This form of notation helps to express such calculations more succinctly and efficiently.

Understanding powers is crucial because it allows us to handle much larger numbers without having to write out exceedingly long multiplication sentences. In essence, learning powers is like learning a shortcut in arithmetic which makes calculations quicker and promotes deeper insight into number patterns and behaviors.

Multiplication

Multiplication is a fundamental arithmetic operation that represents repeated addition. When performing exponentiation as in our example \(5^{4}\), multiplication is the operation we use to compute the number by itself the number of times indicated by the exponent.

Here's why breaking it down into multiplication steps is helpful:

• Start with two numbers: In \(5^{4}\), we start by multiplying the first two 5s to get 25.
• Continue multiplying: Take the result (25) and multiply it by the next 5 (giving 125).
• Repeat: Multiply the result again by the next 5 to arrive at the final product of 625.

These steps demonstrate that multiplication is a building block for the operation of exponentiation, and focusing on one step at a time can simplify complex calculations. Understanding multiplication basics makes working with exponents intuitive.

Evaluating Expressions

Evaluating mathematical expressions involves systematically calculating the value of a given expression, based on operations like addition, subtraction, multiplication, and exponentiation. In the context of evaluating \(5^{4}\), we apply the rules of exponentiation and multiplication to find our result.

Steps to Evaluate Expressions with Powers for any base and exponent will involve:

• Identifying the base and exponent in the expression.
• Expanding the expression using the base repeated according to the exponent.
• Sequentially multiplying the terms to arrive at a final value.

By taking these steps methodically, we systematically break down and solve the expression. This process helps build confidence in solving more complex problems, encouraging a structured approach to mathematical thinking.