Problem 20 - Elementary and Intermediate Algebra

Question

Simplify. $$ 5^{4} $$

Step-by-Step Solution

Answer

625

Step 1: Understand the Problem

The problem requires simplifying the expression involving an exponent: $$ 5^4.

Step 2: Expand the Exponent

When you have an expression like $$ a^b, it means you are multiplying a by itself b times. So here, $$ 5^4 means: $$ 5 \times 5 \times 5 \times 5.

Step 3: Perform the Multiplication

Now multiply the numbers step by step: $$ 5 \times 5 = 25 Then, $$ 25 \times 5 = 125 Finally, $$ 125 \times 5 = 625.

Step 4: Write the Final Answer

After performing all the multiplications, the simplified value of $$ 5^4 is $$ 625.

Key Concepts

Exponentiation

Exponentiation is a math operation where a number (called the base) is multiplied by itself a certain number of times (indicated by the exponent). In the example given, the base is 5, and the exponent is 4. This means that we need to multiply 5 by itself 4 times. Mathematically, it’s represented as: \( 5^4 \).

Understanding exponentiation is important because it simplifies repeated multiplication. For example, instead of writing \( 5 \times 5 \times 5 \times 5 \), we just write \( 5^4 \). This saves time and reduces errors.

It’s also helpful for scientific notation, computing large numbers, and understanding mathematical concepts in higher studies like algebra and calculus.

Multiplication

Multiplication is one of the basic arithmetic operations, and it’s essential for understanding exponentiation. In the given exercise, we perform a series of multiplications to simplify \( 5^4 \):

• Start with 5 and multiply it by 5, which gives you 25
• Then take that result (25) and multiply it by 5 again, resulting in 125
• Finally, multiply 125 by 5 to get 625

So, the multiplication steps are: \( 5 \times 5 = 25 \), followed by \( 25 \times 5 = 125 \), and then \( 125 \times 5 = 625 \). Breaking it into smaller steps makes it easier to understand and manage.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. Understanding these operations is fundamental to learning math, especially when dealing with exponents.

Here’s a quick guide to these operations:

• Addition: Combining two numbers to get a sum. For example, \( 2 + 3 = 5 \).
• Subtraction: Finding the difference between two numbers. For example, \( 5 - 2 = 3 \).
• Multiplication: Adding a number to itself a specified number of times. For example, \( 3 \times 4 = 12 \).
• Division: Splitting a number into equal parts. For example, \( 12 \divide\ 4 = 3 \).

The problem we solved falls under multiplication. By understanding and practicing basic arithmetic operations, you'll find solving more complex problems, like exponentiation, much easier.