Problem 20 - Elementary Algebra

Question

For Problems \(1-20\), find the value of each numerical expression. For example, \(2^{4}=2 \cdot 2 \cdot 2 \cdot 2=16\). $$ -(1.1)^{2} $$

Step-by-Step Solution

Answer

The value is -1.21.

Step 1: Identify the Base and Exponent

In the expression \(-(1.1)^2\), the base is 1.1 and the exponent is 2. This means you need to multiply 1.1 by itself.

Step 2: Calculate the Power

Compute \((1.1)^2 = 1.1 \times 1.1\). Multiply 1.1 by 1.1 to get 1.21.

Step 3: Apply the Negative Sign

Apply the negative sign to the result of the power calculation. Since the original expression is \(-(1.1)^2\), you take \(-1 \times 1.21\) to get -1.21.

Step 4: Conclude the Calculation

The value of the expression \(-(1.1)^2\) is -1.21.

Key Concepts

Base and Exponent

In mathematics, understanding the concepts of base and exponent is crucial for grasping more complex numerical expressions. The base is the number that you are repeatedly multiplying. In the expression \(1.1^2\), the base is 1.1. This tells us we will use 1.1 in a repetitive multiplication process.
The exponent is the small number written above and to the right of the base. It indicates how many times the base is multiplied by itself. In our case, the exponent is 2. Therefore, for the expression \(1.1^2\), we have to multiply 1.1 by itself, as directed by the exponent 2.
Understanding the role of base and exponent helps simplify seemingly complex mathematical expressions into easier calculations by breaking them down into simple multiplication steps.

Power Calculation

Power calculation involves multiplying a number by itself as many times as the exponent indicates. When you see an expression like \(1.1^2\), you perform power calculation by multiplying 1.1 by itself.

• First, write down the base: 1.1.
• Then, apply the exponent, which in this example is 2, meaning you will calculate \(1.1 \times 1.1\).
• Perform the multiplication: 1.1 \(\times\) 1.1 equals 1.21.

In this straightforward manner, power calculation helps convert an exponent expression into a basic arithmetical operation, yielding the result 1.21 for \(1.1^2\). This crucial step simplifies decision-making and ensures accurate results when dealing with powers.

Negative Sign in Expressions

A negative sign in a mathematical expression requires careful attention. It affects the entire calculation and must be applied last to ensure accurate outcomes. In the expression \(- (1.1)^2\), the negative sign is outside the parentheses, affecting the result of the base and exponent calculation.
Here’s how to handle it:

• Perform the power calculation first. As previously solved, \(1.1^2\) equals 1.21.
• Next, apply the negative sign. The expression becomes \(-1 \times 1.21\).
• The final result will be -1.21 after your multiplication.

Thus, a negative sign can transform a positive result into a negative one, making it essential to apply it correctly to avoid errors in your calculations.