Problem 9 - Fundamentals of Mathematics

Question

Determine the value of each power and root. \(2^{5}\)

Step-by-Step Solution

Answer

The value of \(2^5\) is 32.

Step 1: Understand the Exponent

The expression given is \(2^5\). Here, \(2\) is the base and \(5\) is the exponent. This means we need to multiply the base, \(2\), by itself a total of 5 times.

Step 2: Multiply the Base

Start multiplying the base by itself the number of times indicated by the exponent: \[ \begin{align*} 2 imes 2 & = 4 \end{align*} \] This is the result of multiplying \(2\) twice.

Step 3: Continue the Multiplication

Continue multiplying the result from the previous step by \(2\) until you reach the total indicated by the exponent:\[ \begin{align*} 4 imes 2 & = 8 \end{align*} \]

Step 4: Finalize the Calculation

Multiply the result by \(2\) two more times to complete the calculation:\[ \begin{align*} 8 imes 2 & = 16 \ 16 imes 2 & = 32 \end{align*} \]

Step 5: Conclusion

The calculation is complete. The value of \(2^5\) is \(32\).

Key Concepts

Powers

When dealing with powers in mathematics, we often refer to expressions like \(2^5\). A power expression is a way of writing repeated multiplication in a compact form. For instance, instead of writing \(2 \times 2 \times 2 \times 2 \times 2\), we can simply write \(2^5\). Here, the number pushed up (the 5 in \(2^5\)) is called the exponent, while the number multiplied by itself (the 2 in \(2^5\)) is called the base.

This concept is widely useful because it helps simplify calculations and makes expressions more readable. Powers are found not just in math problems, but in real-life applications such as computing compound interest and exponential growth in populations.

To solve powers efficiently:

• Identify the base and the exponent.
• Multiply the base by itself as many times as the exponent shows.

This allows for easier and faster computation, especially when dealing with larger numbers.

Base and Exponent

In mathematics, the terms base and exponent are fundamental when discussing powers. The base is the number that is being multiplied. In the example \(2^5\), the base is 2. It tells us what number we start with in the repeated multiplication process.

The exponent, on the other hand, tells us how many times to multiply the base by itself. A common way to think about it is that the exponent is the number of times you "use" the base in a multiplication sequence. In \(2^5\), the 5 is the exponent, which means that we will multiply the number 2 a total of 5 times.

Understanding the role of the base and the exponent is crucial for calculating powers accurately. Always ensure that you:

• Correctly identify which number is the base.
• Apply the exponent as the count of times the base must be used in multiplication.

This will ensure you correctly interpret and compute any power expression.

Multiplication

Multiplication is the key operation when working with powers. While it might seem straightforward, understanding how multiplication plays into calculating powers is essential.

For example, in \(2^5\), you begin with the base, which is 2. Then, according to the exponent, you multiply 2 by itself 5 times. Here's how it progresses:

• Start: 2
• Step 1: \(2 \times 2 = 4\)
• Step 2: \(4 \times 2 = 8\)
• Step 3: \(8 \times 2 = 16\)
• Step 4: \(16 \times 2 = 32\)

By the end of these steps, you reach the final value of the power expression: \(2^5 = 32\).

Multiplication in powers requires careful attention to ensure every step follows accurately, thus allowing you to reach the correct final result.

Calculation of Exponential Expressions

Calculating exponential expressions may look challenging, but by breaking it down, it becomes manageable and clear.

An exponential expression, such as \(2^5\), tells us to multiply the base number (2) a specific number of times, defined by the exponent (5). The calculation follows these simple steps:

• Identify the base (2) and the exponent (5).
• Multiply the base by itself, keeping track of each step.

Continuing from above indentation, here's the step-by-step:

1. \(2 \times 2 = 4\)
2. \(4 \times 2 = 8\)
3. \(8 \times 2 = 16\)
4. \(16 \times 2 = 32\)

With each step, you get closer to the final result. This systematic approach ensures no mistakes and leads you to the solution. By applying these steps consistently, any exponential expression can be accurately computed.