Problem 49 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-15} \sqrt{-5} $$

Step-by-Step Solution

Answer

\(-5\sqrt{3}\)

Step 1: Express each square root in terms of i

Recall that \ \( \sqrt{-1} = i \). Therefore, any square root of a negative number can be expressed in terms of \(i\). For \(\sqrt{-15}\), express it as: \[ \sqrt{-15} = \sqrt{-1 \times 15} = \sqrt{-1} \times \sqrt{15} = i\sqrt{15} \].Similarly, for \(\sqrt{-5}\):\[ \sqrt{-5} = \sqrt{-1 \times 5} = \sqrt{-1} \times \sqrt{5} = i\sqrt{5} \].

Step 2: Multiply the expressions

Now that both expressions are in terms of \(i\), multiply them together:\[ i\sqrt{15} \times i\sqrt{5} = i^2 \times \sqrt{15} \times \sqrt{5} \].Since \(i^2 = -1\), the expression simplifies to:\[ -1 \times \sqrt{15} \times \sqrt{5}\].

Step 3: Simplify the product of the square roots

Combine \(\sqrt{15}\) and \(\sqrt{5}\) using the property \(\sqrt{a}\times\sqrt{b} = \sqrt{a \times b}\):\[ \sqrt{15} \times \sqrt{5} = \sqrt{15 \times 5} = \sqrt{75}\].

Step 4: Simplify the square root

Simplify \(\sqrt{75}\):Since \(75 = 25 \times 3\),\[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\].

Step 5: Combine and finalize the expression

Substitute back into the expression from Step 2:\[ -1 \times 5\sqrt{3} = -5\sqrt{3}\].Therefore, the simplified expression of \(\sqrt{-15} \sqrt{-5}\) is \(-5\sqrt{3}\).

Key Concepts

Imaginary Unit

The imaginary unit, typically denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the equation \(i^2 = -1\). This definition allows us to extend the real number system to include complex numbers, which are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Understanding the imaginary unit is crucial when dealing with the square roots of negative numbers. Normally, the square root of a negative number is not possible within the real number system. However, by introducing the imaginary unit, these calculations become feasible.

So, when you see an expression like \(\sqrt{-x}\), you can rewrite it in terms of \(i\) by recognizing that \(\sqrt{-1} = i\) and express \(\sqrt{-x}\) as \(i\sqrt{x}\). For example, \(\sqrt{-15}\) can be expressed as \(i\sqrt{15}\). This manipulation allows us to perform arithmetic operations on quantities that include square roots of negative numbers.

Square Roots of Negative Numbers

Traditionally, the square root function is defined for non-negative numbers only. However, when dealing with negative numbers, we need to use the concept of the imaginary unit \(i\) to find the square roots.

To express the square root of a negative number using \(i\), you first factor the number under the square root as \(-1\) times a positive number. Then, apply the property that \(\sqrt{-1} = i\). This transforms expressions like \(\sqrt{-b}\) into \(i\sqrt{b}\).

For example, to find \(\sqrt{-5}\), recognize it as \(\sqrt{-1 \times 5}\), which then becomes \(\sqrt{-1} \times \sqrt{5} = i\sqrt{5}\). This expression fits into the broader concept of complex numbers where the square roots of negative numbers are expressed in terms of the imaginary unit.

Simplification of Radical Expressions

Simplifying radical expressions, especially those involving negative numbers, involves a few steps and the use of properties of radicals. These properties include the ability to multiply and simplify expressions under the radical sign.

When simplifying radicals, like \(\sqrt{15} \times \sqrt{5}\), you can use the multiplication rule for square roots: \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). In our problem, this becomes \(\sqrt{15 \times 5} = \sqrt{75}\).

The next step is to simplify \(\sqrt{75}\) further. Factorize the number 75 as \(25 \times 3\), and use the property that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This results in \(\sqrt{25} \times \sqrt{3}\), which simplifies to \(5\sqrt{3}\), since \(\sqrt{25} = 5\).

By understanding these simplification techniques, you can handle complex expressions involving radicals more easily, ensuring they are expressed in their simplest form.