Problem 1 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(3^{3}\)

Step-by-Step Solution

Answer

The value of the power \(3^3\) is 27.

Step 1: Understanding the Base and Exponent

The given expression is \(3^3\). Here, the base is 3, and the exponent is 3. This means that we need to multiply the base by itself as many times as the exponent indicates. Essentially, this represents 3 multiplied by itself 3 times.

Step 2: Calculating the Product

We start by multiplying 3 by itself. First calculate \(3 \times 3 = 9\).

Step 3: Final Multiplication

Next, take the result from the previous step and multiply by the base again: \(9 \times 3\).

Step 4: Obtaining the Final Result

Calculate \(9 \times 3 = 27\). This tells us that \(3^3 = 27\).

Key Concepts

Base and Exponent

In mathematics, understanding the terms 'base' and 'exponent' is fundamental when dealing with powers. A power is expressed in the format of a base raised to an exponent, such as in the expression \(3^3\). Here, the base is the number 3. The exponent, also 3, indicates the number of times the base is used as a factor in a multiplication operation.
When reading the expression \(3^3\), it can be framed as "3 to the power of 3," meaning you multiply 3 by itself, which is repeated 3 times.

• **Base**: the number to be multiplied (in this case, 3)
• **Exponent**: the number of times the base is multiplied by itself (here, 3)

Recognizing how these components work together helps in simplifying and solving expressions involving exponents effectively.

Multiplication

When working with exponents, multiplication is a key process. It involves multiplying the base by itself the number of times indicated by the exponent. Let’s take a closer look at how this applies to the expression \(3^3\).
The expression requires multiple rounds of multiplication:

• First, multiply the base 3 by itself: \(3 \times 3 = 9\).
• Then, continue by taking the result, which is 9, and multiplying it by the base once more: \(9 \times 3 = 27\).

This step-by-step approach of multiplication helps ensure accuracy and understanding in calculating powers. The process can be summarized as a sequence of multiplication steps where each step builds upon the last result.

Powers and Roots

The concept of powers extends beyond simple multiplication. It opens up a relationship between numbers that involves repeated factors. The expression \(3^3 = 27\) illustrates the power of 3, determined by using 3 as a factor three times. This technique of multiplying the same number repeatedly characterizes what we call 'raising to a power.'
Additionally, there’s a counterpart concept in mathematics known as roots, which can be seen as reversing the role of powers. For instance, finding the cube root of 27 asks: "What number, when used as a factor three times, results in 27?" The answer is 3, which is the base of our original expression.
By understanding powers and roots, one gains the ability to handle both operations involving exponential expressions and their inverses. This balance of computation provides a robust understanding of mathematical relationships.