Problem 26 - Algebra 2

Question

Simplify. $$ \sqrt{-144} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{-144}\) is \(12i\).

Step 1: Identify and Separate the Parts

Identify the expression inside the square root, which is \(-144\). Since we have a negative number inside the square root, recall that \(\sqrt{-1}\) is equal to \(i\), an imaginary unit, so \(\sqrt{-144} = \sqrt{-1 \times 144}\). Now, rewrite it as \(\sqrt{-1}\cdot \sqrt{144}\).

Step 2: Simplify the Square Root of Negative One

Simplify \(\sqrt{-1}\) using the imaginary unit \(i\). Thus, \(\sqrt{-1} = i\).

Step 3: Simplify the Square Root of Positive 144

Calculate the square root of \(144\). Since \(144\) is a perfect square, \(\sqrt{144} = 12\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to express the whole square root. Therefore, \(\sqrt{-144} = i \times 12 = 12i\).

Key Concepts

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. Square roots from positive numbers are straightforward because they yield real numbers. However, when dealing with negative numbers inside a square root, we need to use imaginary numbers. An important point to remember is that not all numbers have real number square roots. Negative numbers do not have real square roots since no real number times itself results in a negative. This is where imaginary numbers come into play. The square root symbol is √, and finding the square root is a common step in solving algebraic equations. Often, you will need to rearrange an equation or manipulate square roots to simplify expressions.

Complex Numbers

Complex numbers are a combination of real numbers and imaginary numbers. They are written in the form of \(a + bi\), where \(a\) is the real part, and \(b\) is the coefficient of the imaginary part \(i\), where \(i = \sqrt{-1}\). The concept of complex numbers extends our understanding of numbers beyond the real number line. This is essential in various fields of science and engineering, especially in solving polynomial equations that have no real solutions.

• Real part: The real part is any number from the real number system.
• Imaginary part: The imaginary part is composed of the imaginary unit \(i\) and a real number.

Complex numbers can be manipulated using algebraic methods, including addition, subtraction, multiplication, and division.

Perfect Squares

Perfect squares are numbers that are made by squaring a whole number. For instance, 1, 4, 9, 16, and so on are perfect squares because they are 1², 2², 3², 4², respectively. Recognizing perfect squares is useful in simplifying square roots and solving equations.

• Identifying perfect squares: These numbers are straightforward as they result from an integer multiplied by itself. For example, 144 is a perfect square because \(12^2 = 144\).
• Simplifying square roots: Knowing perfect squares helps in simplifying square roots efficiently. For example, \(\sqrt{144} = 12\).

Perfect squares show up frequently in algebra, making it helpful to be able to recognize and work with them efficiently.