Problem 2 - Algebra 2

Question

Simplify. $$ \sqrt{(-2)^{2}} $$

Step-by-Step Solution

Answer

2

Step 1: Understand the Problem

The task is to simplify the expression \( \sqrt{(-2)^{2}} \). This involves knowledge about exponents and square roots.

Step 2: Calculate the Exponent

First, focus on the expression inside the square root: \((-2)^{2}\). To calculate this, multiply -2 by itself: \(-2 \times -2 = 4\). Thus, \((-2)^{2} = 4\).

Step 3: Apply the Square Root

Now apply the square root to the result obtained in the previous step: \(\sqrt{4}\). The square root of 4 is 2, since \(2 \times 2 = 4\).

Step 4: Write the Simplified Expression

The expression \( \sqrt{(-2)^{2}} \) simplifies to 2. Therefore, the answer is 2.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication. When you see an expression like \((-2)^2\), it means you multiply \(-2\) by itself. So, in this instance, you perform the operation: \(-2 \times -2\). Exponents are often referred to as powers, and are written as a small number to the top right of a base number.

• The base here is \(-2\).
• The exponent or power is 2.

When multiplying two negative numbers together, such as in our expression, the result is a positive number. That's because the negative signs cancel each other out:

• \(-2 \times -2 = 4\)

Therefore, \((-2)^2 = 4\). Exponents provide a convenient way to simplify expressions and calculate repeated multiplications quickly.

Square Roots

Square roots allow us to find a number which, when multiplied by itself, gives the original number. The square root is denoted by the symbol \(\sqrt{}\). For example, in the expression \(\sqrt{4}\), we want to find a number that gives 4 when multiplied by itself.

• That number is 2, because \(2 \times 2 = 4\).
• The square root of 4 is notated as \(\sqrt{4} = 2\).

Understanding square roots is important when simplifying expressions. They often show up in various areas of math, simplifying equations or expressions for easier evaluation. Remember that a square root "undoes" the squaring of a number, bringing us back to the original base (when it's a perfect square).

Negative Numbers

Negative numbers are less than zero and are denoted by a minus sign (−). They are used to represent values below zero like temperatures, elevations, or when describing debts.When working with negative numbers, especially with operations like multiplication or division, specific rules govern the outcomes:

• When multiplying or dividing two negative numbers, the result is positive.
• When multiplying or dividing a negative number by a positive number, the result is negative.

In our original problem, we have \((-2)^2\), which involves the product of two negative numbers. As we calculated:

• \(-2 \times -2 = 4\), resulting in a positive number because the negative signs cancel each other.

Mastering the rules for negative numbers helps in accurately solving problems that involve a mix of positive and negative values, enabling simplification of expressions such as ones seen in this exercise.