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Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part of a complex number is expressed using the imaginary unit denoted as "i," which is defined by the property that \(i^2 = -1\). This allows us to extend the real number system to include solutions to equations that involve square roots of negative numbers. For example:

\( \sqrt{-1} = i \),
\( \sqrt{-4} = 2i \),
\( \sqrt{-12} = \sqrt{-1} \times \sqrt{12} = i \sqrt{12} \).

So, complex numbers are extremely useful when you encounter an equation like \( p^2 = -12 \). Here, we use the square root property to introduce the imaginary number, allowing us to express the roots as \( p = \pm i\sqrt{12} \). By using complex numbers, we can solve otherwise difficult equations and delve into realms extending beyond just real numbers.

When solving equations involving squares, such as \(3p^2 + 36 = 0\), one often uses the square root property to find solutions. The square root property tells us that if \(a^2 = b\), then \(a = \pm \sqrt{b}\). Here's how it applies:

• First, isolate the term with the square (\(p^2\)) by performing operations like subtraction or addition on both sides of the equation.
• Divide through by any coefficient that multiplies the squared term to isolate it completely.
• Apply the square root property to solve for the unknown variable.

This sequence of steps systematically breaks down the equation into manageable parts, allowing even seemingly daunting problems like \(p^2 = -12\) to be solved using structured logical steps. This logical flow in equation solving helps when handling equations that involve both real and complex solutions.

Simplifying radicals is a crucial step in making expressions easier to understand and compute. A radical expression can often be broken down into simpler components to reveal its base structure. Here's the process using \(\sqrt{12}\) as an example:

• First, identify the factors of the number under the square root. For 12, it's 4 and 3 (since \(12 = 4 \times 3\)).
• Next, simplify \(\sqrt{4 \times 3}\) by separating it into \(\sqrt{4} \times \sqrt{3}\).
• Calculate each of these simpler square roots. \(\sqrt{4} = 2\), so \(\sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).

Applying this simplification to \(\sqrt{-12}\), we insert the imaginary unit as part of our simplification, so \(\sqrt{-12} = i\sqrt{12} = 2i\sqrt{3}\).

Through simplifying radicals, you not only tidy up expressions but also prepare them for further operations or to identify properties such as whether they represent complex numbers.