Problem 48 - Intermediate Algebra

Question

Evaluate each expression. See Example \(6 .\) $$ 2^{5} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

The expression given is \(2^5\). This means that the base \(2\) is raised to the power of \(5\). To evaluate this, you will have to multiply the base \(2\) by itself a total of five times.

Step 2: Start Calculating the Powers

Begin by multiplying the base \(2\) by itself: \(2 \times 2 = 4\).

Step 3: Continue the Multiplication

Now take the result from the previous step and multiply by base \(2\) again: \(4 \times 2 = 8\).

Step 4: Multiply Again

Take the result of the last multiplication and multiply by base \(2\) again: \(8 \times 2 = 16\).

Step 5: Final Multiplication

Finally, take the most recent result and multiply by base \(2\) one last time: \(16 \times 2 = 32\).

Key Concepts

base and exponent

In mathematics, understanding the concept of 'base and exponent' is crucial, especially when dealing with exponentiation. An exponent is a small numeral placed at the upper right of a base number. It indicates how many times the base number is multiplied by itself. For example, in the expression \(2^5\), the number \(2\) is the base, and the number \(5\) is the exponent. This denotes that \(2\) should be multiplied by itself a total of five times.

In general, the operation can be visualized like this:

• The base is the number you start with. It is the number you will multiply.
• The exponent tells you how many of these multiplications you perform.

When you see this notation, simply remember to repeat the base as a factor according to the exponent's value.

power of a number

The term 'power of a number' refers to the result of multiplying a number by itself, as indicated by its exponent. Using our example, \(2^5\), we can say that it represents the "fifth power of 2." Understanding powers is essential because they simplify repeated multiplication into a concise notation. Instead of writing \(2 \times 2 \times 2 \times 2 \times 2\), we simply express it as \(2^5\).

This notation allows mathematicians and students alike to handle complex calculations more efficiently.

• Powers help streamline calculations and solve algebraic problems.
• Each increase in the exponent represents an additional layer of multiplication.

Recognizing powers can dramatically reduce both the time and space needed to write out these operations.

step-by-step calculation

Evaluating expressions, especially those involving powers, can be made straightforward through step-by-step calculations. Let's revisit how \(2^5\) is computed:

1. **Start with the base:** Begin by identifying the base, which is \(2\) in this case.

2. **Multiply successively:**
- First, calculate \(2 \times 2 = 4\).
- Then, multiply the result by \(2\) again: \(4 \times 2 = 8\).
- Once more, multiply the result by the base: \(8 \times 2 = 16\).
- Lastly, multiply your most recent result by \(2\) to yield \(16 \times 2 = 32\).

This systematic approach not only helps in obtaining the correct result but also boosts confidence in dealing with more complex expressions.

• Perform calculations sequentially and verify each step.
• Breaking down operations into smaller tasks prevents errors.

By following this method, calculations become manageable, and any mistakes can be identified and corrected quickly.