Problem 17 - Fundamentals of Mathematics

Question

Find each value. \((-5-6)^{2}\)

Step-by-Step Solution

Answer

The value of \((-5-6)^2\) is 121.

Step 1: Calculate the Sum Inside the Parentheses

Start by solving the expression inside the parentheses. We have \(-5 - 6\). Calculate this by adding the numbers: \(-5 + (-6) = -11\).

Step 2: Square the Result

Now, take the result from the first step, which is \(-11\), and square it: \((-11)^2 = (-11) \times (-11) = 121\).

Key Concepts

Squaring Numbers

Squaring a number means multiplying the number by itself. When you square a number, it is raised to the power of 2, which you will see as the exponent "2" displayed as a small number above and to the right of the base number. For example, squaring 3 is written as \(3^2\), which equals \(3 \times 3\). This process turns a negative multiplied by itself into a positive. So, \((-11)^2\) equates to \((-11) \times (-11)\) and results in a positive 121.

• Squaring always results in a positive number, even if you start with a negative number.
• Powers of 2 are always easier to handle, as you are simply multiplying the number by itself.

It's important to remember that when any number is squared, you're effectively doubling its visual representation, much like turning a line into a square.

Order of Operations

Order of operations is a crucial concept in mathematics to properly solve equations involving multiple steps and operators. A memorable acronym for order of operations is PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

The order of operations dictates that you must resolve parts of the equation inside parentheses first. In our example, calculating \(-5 - 6\) inside the parentheses gives \-11\. After handling parentheses, you proceed to exponents, like squaring the results. This structured approach ensures everyone solves math problems in the same way!
For success, follow PEMDAS and tackle each step methodically to avoid mistakes, especially in complex equations.

Negative Numbers

Negative numbers are numbers less than zero, represented by a minus sign \(-\) before the number. Working with negatives can be tricky, which is why understanding them is key. Here’s what to remember:

• When you add two negative numbers, you "move" further left on the number line, increasing the negative value. For instance, \(-5 + (-6)\) equals \(-11\).
• When you square a negative number, the minus sign "disappears" because two negatives make a positive. For example, \((-11) \times (-11) = 121\).

The concept of negative numbers often centers around the idea of debt or below-zero values. Mastering them is essential to solving many math problems accurately.
By spending time practicing and understanding how to work with negative numbers, you turn potential math difficulties into easy victories!