Problem 5 - Algebra 2

Question

Multiply, if possible. Then simplify. $$ \sqrt{-5} \cdot \sqrt{5} $$

Step-by-Step Solution

Answer

The result of multiplying \(\sqrt{-5}\) by \(\sqrt{5}\) is an imaginary number, which is \(5i\).

Step 1: Understating the imaginary unit

The first thing to understand is that the square root of a negative number can be expressed using the imaginary unit 'i'. The square root of -1 is defined as the imaginary unit 'i', i.e., \(\sqrt{-1} = i\). This is because squaring i gives -1, i.e., \(i^2 = -1\). So, the problem \(\sqrt{-5} \cdot \sqrt{5}\) can be broken down as \(\sqrt{-1} \cdot \sqrt{5} \cdot \sqrt{5}\).

Step 2: Perform Multiplication

Now, multiply \(\sqrt{-1}, \sqrt{5}\), and \(\sqrt{5}\) together. Since, \(\sqrt{-1} = i\), we can substitute \(\sqrt{-1}\) with 'i'. We also know that the multiplication of same square roots is just the number itself, i.e., \(\sqrt{5}*\sqrt{5} = 5\). Combining these, we get \(i \cdot 5\) or \(5i\).

Step 3: Simplification

As the result is an imaginary number, it cannot be further simplified and the final answer is \(5i\).

Key Concepts

Multiplication of Radicals

When multiplying radicals, specifically square roots, the process can often be simplified. A radical is an expression that includes a root, such as a square root. Multiplying radicals follows a unique rule:

• Multiply the values under the radicals together.
• If possible, simplify the resulting expression.

For example, if you have \( \sqrt{a} \cdot \sqrt{b} \), it can be combined as \( \sqrt{a \cdot b} \). This is because the square root of a product can be split into the product of the square roots.

In the given problem, we're multiplying \( \sqrt{-5} \) and \( \sqrt{5} \). Even though \(-5\) is negative, you can still apply this principle because \( \sqrt{-5} \) can be expressed using the imaginary unit 'i.' The operation transforms into a mathematic rearrangement that includes understanding of imaginary numbers.

Simplifying Expressions

Simplification is a key process in mathematics that makes expressions easier to understand or compute. It involves reducing expressions to their simplest form. For radicals, the simplest form is when no perfect squares (other than 1) remain under the square root.

• Combine like terms.
• Factor out any perfect squares.
• When dealing with letters, reduce common factors if present.

In our problem, \( \sqrt{5} \times \sqrt{5} = 5 \) because multiplying a radical by itself eliminates the square root. It turns into just the number under the root. With \( \sqrt{-1} \), which is 'i,' the simplification becomes straightforward: \((-1) \) contributes to the imaginary unit, and the multiplication transforms the expression into \((5i) \).

Thus, the simplification doesn't remove the imaginary number but clearly defines its contribution in the final answer.

Imaginary Unit

The imaginary unit, denoted as 'i', is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This means that the square of 'i' is \(-1\). With this basis,

• The square root of any negative number can be expressed in terms of 'i'.
• Imaginary numbers are results of operating on real numbers with imaginary units.
• Combined with real numbers, they form complex numbers in the form \(a + bi\).

Understanding 'i' is crucial because it allows mathematicians and students to find roots and solutions for equations that don't have real number solutions.

Given \( \sqrt{-5} \), the properties of 'i' turn this into \((i \sqrt{5})\). When multiplied with \(\sqrt{5}\), the result is \(5i\), illustrating the application of 'i' in simplifying expressions with negative roots and making them usable in further mathematical computations.