Problem 20 - Algebra A Combined Function

Question

Evaluate \(-6^{2}\)

Step-by-Step Solution

Answer

The value of \(-6^2\) is \(-36\).

Step 1: Identify the Base and Exponent

In the expression \(-6^2\), the base is \(-6\) and the exponent is \(2\). The exponent indicates that the base should be multiplied by itself.

Step 2: Clarify the Notation

When we see \(-6^2\), it is important to clarify the order of operations. The expression is interpreted as \(-(6^2)\), due to the absence of parentheses around the \(-6\).

Step 3: Calculate the Exponentiation

Compute \(6^2\). This is done by multiplying \(6\) by itself: \(6 \times 6 = 36\).

Step 4: Apply the Negative Sign

Now apply the negative sign. We have \(-(6^2) = -(36) = -36\). This means that \(-6^2 = -36\).

Key Concepts

Understanding the Order of Operations

Mathematics has rules to determine how expressions should be evaluated. These rules are known as the **Order of Operations**. They ensure that everyone solves a math problem the same way, achieving a consistent answer. A common way to remember the order of operations is the acronym **PEMDAS**:

• **P**arentheses: Solve anything inside parentheses first.
• **E**xponents: Address exponents (powers and roots next).
• **M**ultiplication and **D**ivision: Work from left to right.
• **A**ddition and **S**ubtraction: Also from left to right.

In the expression \[-6^2\]no parentheses are explicitly shown around the \(-6\). Because of this, per PEMDAS, the exponent applies only to the \(6\) and not the negative sign. Thus, we interpret it as \(-(6^2)\). This shows how crucial understanding and applying the order of operations is for avoiding mistakes when solving math problems.

Handling Negative Numbers

Negative numbers can be tricky because their behavior isn't always intuitive. A negative number is one that is less than zero, symbolized by a minus sign **(-)**. When dealing with exponents and negative numbers, the placement of parentheses can change the result significantly.
For example, if we write \((-6)^2\), this means \(-6\) is multiplied by itself, resulting in a positive number, \(36\). However, in the expression \[-6^2\], the order of operations indicates the squaring applies only to the 6, not the negative, leading to a result of \(-(6^2) = -36\).
By understanding how negative numbers interact with exponents, we can compute expressions correctly. Always look out for parentheses to determine how an operation applies to the entire expression or part of it.

Exploring Squared Numbers

**Squared numbers** are numbers raised to the power of 2, which means the number is multiplied by itself. It is denoted by the expression \(x^2\). When you see something like \(6^2\), read it as "six squared", meaning \(6 \times 6\).
In our exercise, squaring the number 6 results in 36. Squared numbers frequently occur in algebra and geometry, often used to calculate the area of squares or represent quadratic expressions. They are fundamental in understanding polynomial equations and graphing.
Squaring is a straightforward operation, but like all mathematical operations, it can behave differently with various numbers. Be mindful of whether your base is positive or negative, and watch for parentheses—these little details make a big difference in the outcome of your calculations.