Problem 23 - Algebra A Combined Function

Question

Evaluate \(-0.7^{2}\)

Step-by-Step Solution

Answer

The answer is -0.49.

Step 1: Identify the Base and Exponent

In the expression \(-0.7^{2}\), identify the base as \(-0.7\) and the exponent as 2. The base is the number being multiplied, and the exponent tells how many times to multiply the base by itself.

Step 2: Apply the Order of Operations

According to the order of operations, exponents are solved before multiplication or negation. Therefore, you should evaluate the exponent first. Notice that the number \(-0.7\) is not enclosed in parentheses, so the negative sign will not be involved in squaring.

Step 3: Square the Base

To find the square of \(-0.7\), calculate \((-0.7) \times (-0.7)\). However, since there are no parentheses in \(-0.7^{2}\), the operation is \(-(0.7^{2})\). Therefore, calculate \(0.7 \times 0.7 = 0.49\).

Step 4: Apply the Negative Sign

Now apply the negative sign to the result obtained from squaring the base: \(-0.49\). This is because the negative sign in \(-0.7^{2}\) is outside the exponent.

Key Concepts

Order of Operations

The order of operations is a set of rules that tells you the correct sequence to follow when evaluating a mathematical expression. It's like a recipe that guides you on which steps to take first to get the right answer.
The order typically goes by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), which indicates the priority each operation has. Exponents come after parentheses in this order, which means you solve them right before moving on to multiplication or division.

• If you skip the correct order, the result may be wrong.
• Apply operations in a nested order starting from the inside out, if there are parentheses.
• Always handle exponents first if they appear without parentheses in the problem, just like in this exercise with \(-0.7^{2}\).

Negative Numbers

Negative numbers are numbers less than zero, and they work under different rules than positive numbers. They represent things like debt or temperature below freezing.
When working with negative numbers in arithmetic, it’s important to keep track of the signs.

• Adding a negative number is the same as subtracting its positive counterpart.
• Multiplying or squaring a negative can change its sign, but this depends on whether the negative is included in the base for exponents.

In the example \(-0.7^{2}\), the negative sign is not part of the base being squared because it isn’t enclosed in parentheses. Therefore, the negative sign is applied after the squaring operation.

Squaring Numbers

Squaring a number means multiplying the number by itself. It’s a two-step process that is essential in math when dealing with both geometry and algebra.
For example, in the expression \(x^{2}\), you multiply \(x\) by itself. This is \(x imes x\) .

• If the base is positive, like 0.7, the result is positive.
• When you square a negative number included in parentheses, such as \((-0.7)^2\), it gives a positive result because two negatives multiply to make a positive.
• Without parentheses, as in \(-0.7^{2}\), you square the positive number first then apply the negative sign afterwards.

Understanding the proper way to square numbers both with and without a negative sign is highly important for avoiding mistakes in calculations.