Problem 21 - Algebra 1: Concepts and Skills

Question

Evaluate the power. \(2^{4}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Power

In the power \(2^{4}\), 2 is the base and 4 is the exponent. The exponent tells us how many times to multiply the base (which is 2 in this case) by itself.

Step 2: Perform the Multiplication

Multiply 2 by itself 4 times. It will look like this: 2 * 2 * 2 * 2.

Step 3: Computing the Result

After performing the multiplications, we find that \(2^{4} = 16\).

Key Concepts

Exponents

When we deal with mathematics, exponents are a way to express repeated multiplication of a number by itself. This concept is foundational in many areas of mathematics and is essential to understand. For instance, in the expression \(2^4\), the '4' is the exponent, indicating that the base '2' should be multiplied by itself three additional times (for a total of four times).

It is crucial to recognize that an exponent represents the number of times the base is used as a factor in the multiplication. If the exponent were '3', we'd have \(2 \times 2 \times 2\). With a higher exponent, like '4', as in our example, it becomes \(2 \times 2 \times 2 \times 2\), which equals 16. Exponents are not just limited to small numbers; they can be any integer, and the principle remains the same. Understanding this notion is an invaluable tool for various mathematical concepts, including algebraic operations and functions.

Base and Exponent

Diving deeper into exponents, every power is composed of two parts: the base and the exponent. In our example from the original exercise, \(2^{4}\), '2' is the base, meaning it is the number that gets multiplied repeatedly. The '4' is the exponent showing us the count of multiplications.

It's essential to distinguish the base from the exponent because they have different roles. The base is the factor, the actual number we are considering, and the exponent directs how many times we use the base in the multiplication. This difference is fundamental when solving problems with exponents. For example, if the base changes but the exponent stays the same, the result will differ significantly because a different number is being multiplied each time. Conversely, if the exponent changes, the number of multiplications adjusts, affecting the overall result. Recognizing the interplay between these two components is key to mastering the evaluation of powers.

Multiplication of Powers

Moving on to the guidelines of multiplying powers, we need to understand some rules that make these operations easier to handle. When you have the same base and are multiplying the powers, you can simplify the process by adding the exponents. This principle is especially handy when dealing with larger numbers or when variables are involved.

For example, multiplying \(2^3\) by \(2^2\) results in \(2^{3+2} = 2^5\), which ultimately simplifies down to 32. This same rule applies to any base and to any positive integer exponents. However, when you are multiplying powers with different bases, the process is not as direct, and you'll have to perform the multiplication of the bases for each respective exponent before combining the results. It's a simple but powerful rule that helps manage complex expressions with ease and is a foundational part of algebra.