Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Write each number in simplest form, without a negative radicand. $$\sqrt{-100}$$

Step-by-Step Solution

Answer

The simplest form is \( 10i \).

Step 1: Convert to Complex Number

Since the radicand is negative, use the property of square roots for negative numbers: \( \sqrt{-a} = i\sqrt{a} \) where \( i \) is the imaginary unit. Start by rewriting \( \sqrt{-100} \) as \( \sqrt{-1 \times 100} \), which simplifies to \( i \sqrt{100} \).

Step 2: Simplify the Square Root

Evaluate \( \sqrt{100} \). Since 100 is a perfect square, \( \sqrt{100} = 10 \). Thus, \( i \sqrt{100} = 10i \).

Step 3: Verify the Solution

Verify that the solution is correctly expressed in its simplest form without a negative radicand by ensuring that the imaginary unit \( i \) is used. The expression \( 10i \) meets the criteria.

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the symbol \( i \), is a fundamental concept when dealing with complex numbers, especially when you encounter square roots of negative numbers. Normally, the square root of a number is only defined for non-negative numbers, as negative numbers do not have a real number as their square root in the real number system. This is where the imaginary unit \( i \) comes in handy. It is defined so that \( i^2 = -1 \). This unique property allows us to work with the square roots of negative numbers by effectively 'removing' the negative sign from under the radical.

For instance, when faced with \( \sqrt{-1} \), it is simplified to \( i \). By extending this property, any negative radicand, say \( -a \), can be expressed as \( i\sqrt{a} \), where \( a \) is a positive number.

Square Root Simplification

Square root simplification is an essential mathematical technique employed to express numbers in their simplest radical form. When simplifying a square root, your goal is to extract any perfect square factors from under the radical, leaving the square root of the smallest possible number.

In the problem \( \sqrt{-100} \), after factoring out the negative using the imaginary unit, we are left with \( \sqrt{100} \). Recognizing that 100 is a perfect square (since \( 10^2 = 100 \)), we simplify \( \sqrt{100} \) to 10. Thus, the expression \( i\sqrt{100} \) simplifies to \( 10i \).

This method of simplification ensures that numbers are expressed in a neater form, which makes them easier to handle in further mathematical calculations.

Negative Radicand

The term 'negative radicand' describes a situation where the number inside a square root is negative. This is problematic in traditional real number operations because a square root is not naturally defined for negative numbers. To work around this, mathematicians use the concept of the imaginary unit \( i \).

When tackling \( \sqrt{-100} \), understanding the role of the negative radicand is pivotal. We rewrite it using the property \( \sqrt{-a} = i\sqrt{a} \); therefore, \( \sqrt{-100} = i \sqrt{100} \). By converting the negative radicand into a product involving \( i \), we effectively handle the negativity and transform it into a form that can be manipulated using complex numbers.

The ability to deal with negative radicands opens the door to a broader set of solutions in algebra and beyond, demonstrating the flexibility and utility of complex numbers.