Problem 50 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$(-2.1)^{2}$$

Step-by-Step Solution

Answer

\((-2.1)^2 = 4.41\)

Step 1: Understand the Problem

We need to expand and simplify the expression \((-2.1)^2\). This involves exponentiation, where the base \(-2.1\) needs to be multiplied by itself.

Step 2: Apply the Exponentiation Rule

Exponentiation involves multiplying the base by itself as many times as indicated by the exponent. In this case, \((-2.1)^2\) implies multiplying \(-2.1\) by itself once:\[ (-2.1)^2 = (-2.1) imes (-2.1) \]

Step 3: Perform the Multiplication

Now we calculate the product of \(-2.1\) with itself:\[ (-2.1) imes (-2.1) = 4.41 \]The negative sign multiplied with negative sign becomes a positive sign. Thus, the product is positive.

Step 4: Write the Final Answer

The simplification of \((-2.1)^2\) results in \(4.41\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. It's a way of expressing repeated multiplication of the base. When you see an expression like \((-2.1)^2\), it means that you need to multiply \(-2.1\) by itself.
- The base is the number being multiplied. In our example, this is \-2.1\.- The exponent tells us how many times to multiply the base by itself. Here, the exponent is \2\.
So, \((-2.1)^2\) doesn't mean twice \-2.1\ — it means \-2.1\ times \-2.1\. This will give us the result after performing the multiplication. If you have an exponent of \2\, it's referred to as "squared," indicating two groups of the base multiplied.

Negative Numbers

Negative numbers might seem a bit tricky initially, but they follow straightforward rules during multiplication. When multiplying two negative numbers, the result is positive. This is because the negative signs cancel each other out.
Consider \((-2.1) \times (-2.1)\). Each negative cancels out, converting the product into a positive result.

• Negative \times\ Negative = Positive
• Negative \times\ Positive = Negative
• Positive \times\ Positive = Positive

In the context of exponentiation, ensuring understanding of how negatives interact through multiplication is crucial for obtaining the correct result.

Simplification

Simplification is the process of transforming an expression into its most straightforward form. It's a method of following through calculations to present them more simply.
In the case of our example, we started with \((-2.1)^2\). After performing the multiplication \((-2.1) \times (-2.1)\), we get \4.41\).
- The negative signs were handled, and \(-2.1) \times (-2.1)\ results in a positive product.- Simplification eliminates confusion and makes it easier to work with numbers or expressions.
When you simplify correctly, you ensure that the expression is presented in the most understandable format, ready for any further calculations or interpretations needed.