Problem 71 - Fundamentals of Mathematics

Question

Find the value of \(2^{4}\).

Step-by-Step Solution

Answer

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

Step 1: Understanding the Exponent

The expression we have is \(2^4\). Here, \(2\) is the base and \(4\) is the exponent. The exponent tells us how many times to multiply the base by itself.

Step 2: Expanding the Expression

To solve \(2^4\), multiply the base \(2\) by itself as many times as the exponent indicates. So, we have: \(2 \times 2 \times 2 \times 2\).

Step 3: Performing the Multiplication

Calculate the product step by step: 1. \(2 \times 2 = 4\)2. \(4 \times 2 = 8\)3. \(8 \times 2 = 16\)

Step 4: Conclusion

Thus, the value of \(2^4\) is \(16\).

Key Concepts

Base and Exponent

In mathematics, exponents are used to express repeated multiplication of a number by itself. To understand this, we need to look at two components: the base and the exponent.
The **base** is the number that is being multiplied. In the expression \(2^4\), the number \(2\) is the base. It is the main number that you start with in the multiplication process.
The **exponent**, on the other hand, tells us how many times the base is used in the multiplication. In \(2^4\), the exponent is \(4\), which means you need to multiply the base, \(2\), by itself four times. Thus, the expression \(2^4\) translates to \(2 \times 2 \times 2 \times 2\).
This way of writing and calculating makes it easier to handle and understand expressions involving large numbers and repeated calculations.

Multiplication

Multiplication in the context of exponents involves repeated multiplication of the base. It simplifies what would otherwise be a lengthy addition process.
Imagine you have \(2^4\). This means you need to multiply \(2\) by itself four times. Here are the steps broken down:

• First step: Multiply \(2\) by \(2\), which is \(4\).
• Second step: Multiply the result (\(4\)) by \(2\) again, giving \(8\).
• Third step: Multiply \(8\) by \(2\), resulting in \(16\).

Each of these steps involves straightforward multiplication, and following them helps keep the process organized. Multiplication with exponents is very methodical, making it easier to work with large sets of data and large calculations.

Powers of Numbers

The concept of powers of numbers is central to exponents and describes how numbers are raised to a particular power to simplify expressions.The power of a number indicates how many times the number is used as a factor in the multiplication process. For example, the power of \(2\) in \(2^4\) is \(4\), signifying that the number \(2\) is used as a factor four times.
Understanding powers is crucial because it helps in identifying patterns in multiplication and allows for simplifying complex problems. Powers can also help when comparing different quantities of growth, such as exponential growth in populations or the compounding of interest in finance.
In computational contexts, working with powers and exponents becomes more efficient, reducing time and effort in calculations.