Problem 7 - Prealgebra

Question

Evaluate each expression. $$2^{4}$$

Step-by-Step Solution

Answer

The value of \(2^4\) is 16.

Step 1: Understand the Base and Exponent

In the expression \(2^4\), the number 2 is the base, and 4 is the exponent. The exponent indicates how many times the base is used as a factor.

Step 2: Express Repeated Multiplication

Write the expression as a repeated multiplication. In this case, \(2^4\) means you multiply 2 by itself 4 times: \(2 \times 2 \times 2 \times 2\).

Step 3: Perform the Multiplication

Now, calculate the result of the multiplication:1. First, calculate \(2 \times 2 = 4\).2. Then multiply the result by 2: \(4 \times 2 = 8\).3. Finally, multiply the result by 2 again: \(8 \times 2 = 16\).

Key Concepts

Base and Exponent

When dealing with exponents, understanding the roles of the base and the exponent is crucial. The base is the number that is repeatedly multiplied, and the exponent tells us how many times the base is multiplied by itself.
Think of it as a shortcut for multiplication. Instead of writing 2 multiplied by itself 4 times, we shorten it to \(2^4\).
This makes expressions neater and easier to write, especially for larger numbers. When you see an expression like \(5^3\), 5 is the base, and it is multiplied together 3 times.

• Base: The main number being multiplied
• Exponent: Indicates the number of times the base is used in the multiplication

Repeated Multiplication

Exponents turn repeated multiplication into a simple expression. The exponent in \(2^4\) indicates that the base 2 should be multiplied by itself a total of 4 times. This looks like \(2 \times 2 \times 2 \times 2\).
By breaking it down into steps:

• First, multiply 2 by 2 to get 4.
• Next, take the result 4 and multiply it by 2 to get 8.
• Finally, multiply 8 by 2 to arrive at 16.

This process demonstrates how exponents simplify expressions that involve repeated multiplication, making calculations more efficient.

Prealgebra

In prealgebra, exponents are introduced as a way to simplify repeated multiplication and help students understand more complex mathematical concepts.
This foundational concept prepares you for algebra and beyond by teaching the relationships between numbers in a concise manner. Understanding how to work with exponents helps in grasping the larger picture of math, like polynomial expressions or solving equations.
Learning how exponents work offers several benefits:

• They help solve real-world problems where quantities multiply exponentially, like population growth or compound interest.
• Exponents make it easier to work with very large or very small numbers, such as in scientific notation.

Prealgebra lays the groundwork for these logical ways of thinking, setting you up for future success in mathematics.