Problem 72 - Algebra A Combined Function

Question

Simplify each expression. $$ (-9)^{2} $$

Step-by-Step Solution

Answer

The result is 81.

Step 1: Understand the Expression

The expression \((-9)^{2}\)means that the number \(-9\) is raised to the power of 2, or squared.

Step 2: Apply the Squaring Rule

To square a number means to multiply the number by itself. For the expression \((-9)^{2}\),we calculate \((-9) \times (-9) \).

Step 3: Multiply the Numbers

Now perform the multiplication: \((-9) \times (-9) = 81\).Multiplying two negative numbers results in a positive product.

Key Concepts

Squaring

Squaring is a fundamental operation in mathematics where you multiply a number by itself. It’s like adding a dimension to the original number or seeing how much area a certain length would cover if both sides were equal. Anything raised to the second power is squared.
When you square a number, follow this simple process:

• Take the original number.
• Multiply it by itself.

For example, if you have the expression \((-9)^2\), it involves \(-9\) times \(-9\). This means squaring involves calculative symmetry about the origin, creating a seamless doubling effect that's essential in numerous practical applications like finding areas and understanding parabolic paths in physics.

Negative Numbers

Negative numbers can be a bit tricky, especially when involved in operations like squaring. A negative number is any number less than zero, commonly indicated by a minus sign \((-\)).
This sign drastically changes how it behaves in arithmetic operations.

• When you multiply two negative numbers, the negatives cancel each other out, leading to a positive result.
• This is because a negative action reversed twice becomes a positive action.

Understanding how negative numbers work, especially when squared, is crucial. For instance, when you take \(-9\) and square it, \((-9)^2=81\) shows that the negatives turn into a positive product. Raising a negative number to an even power always results in a positive outcome.

Multiplication

Multiplication is a shortcut for addition. It tells you how many times to add a number to itself. Multiplication becomes intuitive when you relate it to repeated addition. For instance, \(3 \times 4\) means adding \(3+3+3+3\).
When multiplying, it's important to pay attention to the signs of the numbers involved, as they affect the result:

• Positive \(\times\) Positive = Positive: \(2 \times 2 = 4\).
• Negative \(\times\) Negative = Positive: \(-9 \times -9 = 81\), as we saw earlier.
• Positive \(\times\) Negative = Negative: \(3 \times -2 = -6\).

Recognizing these patterns makes multiplying more straightforward. Multiplication involving negative numbers is especially important as it appears in many areas, including solving algebraic expressions and calculating distances when directions change.