Problem 37 - Algebra 1

Question

Evaluate the expression. $$2.5 \cdot 0.5^{2} \div 5$$

Step-by-Step Solution

Answer

The evaluation of the given expression results in 0.125.

Step 1: Exponent Calculation

Firstly, perform the operation of exponentiation in 0.5^{2}. This translates to 0.5 multiplied by itself. Therefore, 0.5^{2} = 0.25.

Step 2: Multiplication

Next, multiply the result of the exponentiation, which is 0.25 by 2.5. The calculation is 2.5 * 0.25 = 0.625.

Step 3: Division

Finally, divide the result from the multiplication by 5. The calculation is 0.625 / 5 = 0.125.

Key Concepts

Exponentiation

One of the foundational elements of algebra is understanding exponentiation. This concept is about raising numbers to powers, which essentially tells you how many times to multiply a number by itself. For example, an expression like \(0.5^2\) involves squaring the number \(0.5\).

Let's break it down: \(0.5^2\) means \(0.5 \times 0.5\), which equals \(0.25\). It is crucial to perform the exponentiation first due to the order of operations, which we'll explore more later.

When working with exponents, remember that the exponent applies only to the number directly before it. So in \(2.5 \cdot 0.5^2\), the exponent \(2\) only affects the \(0.5\), not the \(2.5\).

Multiplication and Division

Once the exponentiation is clear, the next steps usually involve multiplication or division. These operations are on the same level according to the order of operations, meaning you would approach them from left to right as they appear in the problem.

For instance, in our expression after exponentiation, we have the multiplication of \(2.5\) and \(0.25\) to tackle. Multiplying decimal numbers follows the same rules as whole numbers: line up the decimal points and multiply as usual. Here, \(2.5 \times 0.25 = 0.625\).

After multiplication, division comes into play. You would divide \(0.625\) by \(5\) to find the final answer. Division works inversely to multiplication, and with decimals, it is often helpful to consider moving the decimal to create whole numbers, if necessary, to simplify the process.

Order of Operations

The order of operations is a fundamental concept in algebra that establishes the correct sequence to evaluate expressions. It is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This rule dictates that we first address any calculations inside parentheses, then apply exponents, followed by any multiplication or division, and finally addition or subtraction if they are present.

In our example, the order of operations tells us to first calculate the exponent (Step 1), then to multiply (Step 2), and finish with division (Step 3). The strict following of this order ensures that regardless of who calculates the expression, or where in the world they are, they will come up with the same correct answer of \(0.125\).