Problem 12 - Elementary and Intermediate Algebra

Question

Express in terms of \(i\) $$ \sqrt{-19} $$

Step-by-Step Solution

Answer

i\sqrt{19}

Step 1: Understand the Number

Identify that the expression involves a square root of a negative number. The square root of any negative number can be expressed in terms of the imaginary unit, denoted as \(i\), where \(i = \sqrt{-1}\).

Step 2: Rewrite the Negative Number

Express \(-19\) as a product of \(-1\) and 19: \(-19 = -1 \cdot 19\).

Step 3: Apply the Square Root

Use the property of square roots that states \(\sqrt{a\cdot b} = \sqrt{a} \cdot \sqrt{b}\). Therefore, \(\sqrt{-19} = \sqrt{-1 \cdot 19} = \sqrt{-1} \cdot \sqrt{19}\).

Step 4: Express in Terms of \(i\)

Since \(\sqrt{-1} = i\), substitute \(i\) into the expression: \(\sqrt{-1} \cdot \sqrt{19} = i\sqrt{19}\).

Key Concepts

square root

The concept of the square root is fundamental in mathematics. A square root of a number is a value which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 x 3 equals 9. We denote the square root using the radical symbol: \(\backslash sqrt{}\). Understanding square roots helps in simplifying and solving mathematical expressions.

When you see \(\backslash sqrt{-19}\), it might look intimidating due to the negative number under the square root. Don't worry, this is where imaginary numbers come in handy!

negative number

Negative numbers are values less than zero. They are crucial in understanding mathematical concepts involving opposites, deficits, and direction.

In the expression \(\backslash sqrt{-19}\), the negative sign under the square root makes direct simplification tricky, as there is no real number whose square is negative. Enter the imaginary unit, which helps us manage such cases.

imaginary unit

The imaginary unit, denoted as \i\, is an essential concept in complex numbers. It is defined as \(\sqrt{-1}\). With the aid of \i\, we can represent and simplify the roots of negative numbers.

Let's apply this to the expression \(\sqrt{-19}\).

• First, identify the negative number under the square root.
• Next, express the negative number as a product: \(\-19 = -1 \cdot 19\).
• Then break it down using the property of square roots: \(\sqrt{-1 \cdot 19}\).
• We know that \(\sqrt{-1} = i\), so substitute: \(\sqrt{-1 \cdot 19} = \sqrt{-1} \cdot \sqrt{19}\).
• Finally, write it in terms of \i\: \(i\sqrt{19}\).
  

So, \(\sqrt{-19}\) can be expressed as \(i\sqrt{19}\). This makes dealing with the square roots of negative numbers manageable and opens the door to more complex problem-solving.