Problem 6 - Algebra and Trigonometry

Question

Evaluate each expression indicate that the root is not a real number. $$ \sqrt{-25} $$

Step-by-Step Solution

Answer

The result of \(\sqrt{-25}\) is \(5i\).

Step 1: Identify the number under the square root

The number under the square root is -25.

Step 2: Utilize the formula for square root of negative numbers

We know that \(\sqrt{-a}\) equals to \(i\sqrt{a}\). So, we apply this rule here, and say that \(\sqrt{-25} = i\sqrt{25}\).

Step 3: Simplify the square root

The square root of 25 is 5. After simplifying, the result would be \(5i\).

Key Concepts

Understanding Square Roots

Square roots are essential tools in mathematics that allow us to find a number which, when multiplied by itself, gives us the original number. For example, the square root of 25 is 5 because 5 times 5 equals 25. Square roots help us understand and work with quadratic equations, real-life geometry, and much more.
When you see the symbol \(\sqrt{}\), it asks, "What number multiplied by itself equals the number inside?"
However, things change a little when the number under the square root is negative, leading us into the world of imaginary numbers.

The Imaginary Unit

When dealing with negative numbers under a square root, we introduce a concept called the "imaginary unit," denoted by \(i\).
This is because there is no real number that, when squared, gives a negative number. The imaginary unit \(i\) is defined as:

• \(i = \sqrt{-1}\)

By using \(i\), we can extend the set of real numbers to include complex numbers, allowing us to evaluate expressions like \(\sqrt{-25}\).
This way, the solution becomes \(i\sqrt{25}\), leading us to further simplification.

Working with Negative Numbers

Negative numbers can be tricky, especially under a square root. The square root of a negative number initially seems impossible if we only think in terms of real numbers.
But thanks to the imaginary unit \(i\), we can express square roots of negative numbers as complex numbers. For example:

• \(\sqrt{-25} = i\sqrt{25}\)
• This is simplified to \(5i\) since \(\sqrt{25} = 5\)

This transformation lets us work with numbers that don't have real solutions and enables a broader range of mathematical explorations.

Simplification of Radicals

Simplifying radicals is about making expressions easier to interpret. When you have a square root like \(\sqrt{25}\), you simplify by finding the exact number, which is 5.
When combined with the imaginary unit, you follow a similar process:

• Convert \(\sqrt{-a}\) to \(i\sqrt{a}\)
• Simplify \(\sqrt{a}\) to its simplest form, if possible

In our example, \(\sqrt{-25}\) becomes \(i\sqrt{25}\).
This further simplifies to \(5i\), making it less complex and giving you a clear answer.