Problem 40 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$2^{5}$$

Step-by-Step Solution

Answer

The expanded and simplified form of \(2^5\) is 32.

Step 1: Understand the Expression

The expression \(2^5\) means that the base, 2, is raised to the power of 5. This implies that 2 will be multiplied by itself a total of 5 times.

Step 2: Expand the Expression

We can rewrite \(2^5\) as \(2 \times 2 \times 2 \times 2 \times 2\). This means we're multiplying 2 by itself repeatedly, 5 times.

Step 3: Simplify the Expanded Expression

Now, multiply the numbers: \(2 \times 2 = 4\), then \(4 \times 2 = 8\), then \(8 \times 2 = 16\), and finally \(16 \times 2 = 32\).

Key Concepts

Powers of a Number

When a number is raised to a power, like in the expression \(2^5\), it means that we are multiplying the base number by itself a certain number of times. In this case, 2 is the base, and the exponent is 5. The exponent tells us how many times the base number is used as a factor in the multiplication.

• The base is the number that is being multiplied.
• The exponent (or power) tells us how many times the base is multiplied by itself.

For example, in \(2^5\), we multiply 2 a total of 5 times, resulting in \(2 \times 2 \times 2 \times 2 \times 2\). Computing this gives us the final result, 32. This process of repeated multiplication is what we refer to as powers of a number.

Simplification

Simplification is the process of condensing a mathematical expression down to its simplest form. It's like tidying up a messy equation. You start with something more complex and end up with something much simpler.

In the case of \(2^5\), we start by expanding it into its full form: \(2 \times 2 \times 2 \times 2 \times 2\). After that, we carry out the multiplication step-by-step:

• First, \(2 \times 2 = 4\).
• Next, we multiply this result by 2 to get 8.
• Continuing, multiply 8 by 2 to get 16.
• Finally, 16 multiplied by 2 gives 32.

This step-by-step approach ensures that we break down the operation into manageable parts, making it easier to understand and verify each step.

Mathematical Expressions

Mathematical expressions are like sentences made out of numbers, operations, and variables. They convey a particular value or relationship that we can work with. Understanding mathematical expressions involves recognizing components like numbers and operations, as well as knowing how to interpret them.

Consider the expression \(2^5\). It combines the number 2, the exponent 5, and the operation that binds them, which is repeated multiplication. Here's how mathematical expressions work:

• They consist of numbers, variables, and operators (like addition, subtraction, multiplication, division).
• They define a certain computation or relationship.
• They can be simplified or expanded, as shown with \(2^5\).

Expressions like these form the basis for algebraic operations and help in solving equations. By practicing with such expressions, you build the skills to analyze and solve more complex problems.