Problem 32 - Elementary and Intermediate Algebra

Question

$$ -\sqrt{-49} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-7i\).

Step 1: Analyze the Expression

The given expression is \(-\sqrt{-49}\). It is important to notice that the expression involves the square root of a negative number, which has no real number solution.

Step 2: Identify the Imaginary Component

The square root of a negative number involves the imaginary unit \(i\). This is defined as \(i = \sqrt{-1}\). Therefore, to find the square root of \(-49\), it can be expressed with \(i\) as part of the solution.

Step 3: Simplify the Square Root

Rewrite \(-49\) as \(-1 \times 49\). The square root of \(-49\) is \(\sqrt{-1 \times 49}\). This can be rewritten as \(\sqrt{-1} \times \sqrt{49}\).

Step 4: Solve for the Real and Imaginary Components

We know \(\sqrt{-1} = i\) and \(\sqrt{49} = 7\). Therefore, each part of the expression can be solved individually to get \(i \times 7 = 7i\).

Step 5: Apply the Negative Sign

The original expression had a negative sign, so multiply \(7i\) by \(-1\) to obtain \(-7i\).

Key Concepts

Imaginary Unit

Complex numbers are numbers that can have both real and imaginary components. To understand them better, let's first delve into the concept of the imaginary unit, denoted as \(i\). The imaginary unit is defined as the square root of \(-1\). This is a crucial concept because traditional mathematics does not allow square roots of negative numbers within the real numbers system. Therefore, \(i = \sqrt{-1}\).
You can think of \(i\) as a tool or a new number that helps us work with roots of negative numbers.

• \(i^2 = -1\) is a foundational identity of the imaginary unit.
• Extending this, \(i^3 = i \times i^2 = i \times (-1) = -i\) and \(i^4 = (i^2) \times (i^2) = 1\).

Understanding \(i\) assists us in addressing and manipulating numbers that are not possible to solve within the standard realm of real numbers.

Square Root of Negative Number

A common challenge in mathematics is how to deal with the square root of a negative number. In the expression \(-\sqrt{-49}\), we need a way to simplify \(\sqrt{-49}\). The key lies in separating the negative part from the positive part of the square root.
By recognizing that \(-49\) can be expressed as \(-1 \times 49\), we can then split this into two separate square roots: \(\sqrt{-1} \times \sqrt{49}\).

• The first part, \(\sqrt{-1}\), is simplified to \(i\), the imaginary unit.
• The second part, \(\sqrt{49}\), equals \(7\) because \(49\) is a perfect square.

Therefore, the square root of \(-49\) is \(7i\). By understanding these steps, we can compute square roots of negative numbers using the imaginary unit.

Real and Imaginary Components

Complex numbers combine real numbers and imaginary numbers. In our expression, \(-7i\), we have a complex number where the real component is absent and the imaginary component carries the value.

• A complex number generally takes the form \(a + bi\), where \(a\) represents the real part and \(b\) corresponds to the imaginary part.
• In the case of \(-7i\), the real component \(a\) is \(0\), while \(b\), which multiplies \(i\), is \(-7\).

It's essential to separate these two components to truly appreciate complex numbers. While not present in all expressions, understanding where the real numbers and imaginary numbers lie is crucial for solving equations and understanding complex numbers more deeply.