Problem 15 - Algebra 2

Question

Simplify. $$ \sqrt{-(-7)^{2}} $$

Step-by-Step Solution

Answer

The simplified form is \(7i\).

Step 1: Simplify the Expression Inside the Square Root

First, evaluate the expression inside the square root. The expression is \[-(-7)^2\]Here, you have \[(-7)^2\]which means you should square \(-7\). This results in:\[(-7) \times (-7) = 49\]Thus, the expression becomes:\[-(49)\]

Step 2: Evaluate the Negative Outside the Squared Term

Now that we have \[-(49)\]evaluate the negative sign. This turns the expression into \[-49\]The expression inside the square root is now \(-49\).

Step 3: Understand the Square Root of a Negative Number

Recognize that the expression under the square root, \(-49\), is a negative number. The square root of a negative number introduces an imaginary unit "i". Recall that \[i^2 = -1\]. Thus, \[\sqrt{-49} = \sqrt{49} \times \sqrt{-1} = 7i\]

Step 4: Conclusion of the Simplification

After executing the transformation mentioned in Step 3, the simplified form of the expression \[\sqrt{-(-7)^2}\] is \[7i\]. This is the final simplified form of the original expression.

Key Concepts

Imaginary Unit

In mathematics, the concept of imaginary numbers is introduced when dealing with the square root of negative numbers, which leads us to the imaginary unit "i". The defining feature of the imaginary unit is its relation to squares:

• By definition, the imaginary unit "i" is used to express the square root of -1, meaning that \( i^2 = -1 \).
• This unit helps in extending the real number system to include solutions to equations that would otherwise have no real solutions, such as \( x^2 = -1 \).

Think of "i" as a tool that expands our number system, allowing us to express quantities that include the square roots of negative numbers. This abstraction is crucial in many areas of mathematics and engineering where real-world phenomena are captured that real numbers alone cannot express.
In our exercise, the imaginary unit appears when taking \( \sqrt{-49} \), which transforms into \( 7i \) by separating the negative under the square root as explained in the solution process.

Square Root of Negative Number

Taking the square root of a negative number is not something we can do within the realm of real numbers alone, so we introduce the imaginary unit to handle such cases. Typically, when you see a negative number under a square root, you need to think about how to express it using imaginary numbers:

• Identify the negative number under the square root and separate it as the product of a positive number and -1.
• Utilize the identity \( \sqrt{-1} = i \) to handle the -1 part.
• For example, given \( \sqrt{-49} \), recognize it as \( \sqrt{49 \times (-1)} \).
• This further simplifies to \( \sqrt{49} \times \sqrt{-1} \), resulting in \( 7i \) since \( \sqrt{49} = 7 \).

This approach simplifies the handling of square roots of negative numbers, offering solutions that are rooted in the expanded complex number system. This expanded system incorporates all real numbers and imaginary numbers such as "i".

Simplifying Expressions

Simplifying mathematical expressions is crucial in making calculations more manageable. This is especially true when dealing with complex numbers involving imaginary units. Let's walk through the key steps to simplify an expression with a square root of a negative number:

• Evaluate the Expression Inside: Begin by working on any squared terms, as they are often straightforward solutions. For example, \((-7)^2 = 49\).
• Apply Negation: Apply any negative signs outside the squared result. In our example, this means applying the negative to convert \(49\) into \(-49\).
• Handle the Square Root of the Negative: Use the property of imaginary numbers to separate the real and imaginary parts, like \( \sqrt{-49} \) becoming \( 7i \).

Following these steps ensures that expressions are simplified correctly by handling real and imaginary components separately and logically. This careful step-by-step process makes complex calculations much more approachable.