Problem 47 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-9} \sqrt{-6}$$

Step-by-Step Solution

Answer

The solution is \\(-3\sqrt{6}\\).

Step 1: Convert Negative Square Roots to Complex Numbers

Recognize that both numbers under the square root are negative. To handle this, express \(-9\) and \(-6\) with the imaginary unit \(i\). Thus, \(\sqrt{-9} = i\sqrt{9}\) and \(\sqrt{-6} = i\sqrt{6}\).

Step 2: Multiply the Terms

Multiply the expressions obtained in the first step: \(\sqrt{-9} \sqrt{-6} = (i\sqrt{9})(i\sqrt{6})\). This equals \(i^2 \sqrt{54}\), using the property that \(i \cdot i = i^2\).

Step 3: Simplify Using Properties of Imaginary Unit

Recall that \(i^2 = -1\). Substitute to get the expression \(i^2 \sqrt{54} = -\sqrt{54}\).

Step 4: Simplify the Square Root

Further simplify \(\sqrt{54}\) to its simplest form by factorizing: \(\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6}\).

Step 5: Combine and Conclude

Substitute the simplified square root back into the expression to obtain \(\sqrt{-9} \sqrt{-6} = -3\sqrt{6}\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the field of complex numbers. It is defined as the square root of \(-1\). This means that \(i^2 = -1\). The introduction of \(i\) allows us to work with square roots of negative numbers, which are not possible within the real number system alone.

• When encountering a negative number under a square root, we can express it as an imaginary number. For instance, \(\sqrt{-9}\) becomes \(i\sqrt{9}\), which simplifies to \(3i\).
• Imaginary numbers can be combined with real numbers to form complex numbers, written as \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part.

Understanding and using the imaginary unit \(i\) is crucial when dealing with equations that involve square roots of negative numbers, enabling these expressions to be simplified and manipulated within the realm of complex numbers.

Square Roots

Square roots are an essential concept in mathematics, representing a value that, when multiplied by itself, gives the original number. The square root of a positive number \(x\) is denoted as \(\sqrt{x}\).

When it comes to negative numbers, the direct computation of their square roots involves the use of the imaginary unit \(i\). For example, \(\sqrt{-6}\) can be expressed as \(i\sqrt{6}\).

• This transformation is necessary because the square root of a negative number doesn't exist within the set of real numbers.
• Instead, we use the property of the imaginary unit to rewrite these expressions, enabling their use in equations and simplifying operations, especially in the context of complex numbers.
• The process of simplifying square roots often involves factorization. For example, \(\sqrt{54}\) can be broken down into \(\sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6}\).

By mastering these transformations and simplifications, you can effectively solve expressions involving square roots, even when negative numbers are present under the radical.

Multiplication of Complex Numbers

Multiplying complex numbers involves combining both the real and imaginary components. The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

To multiply two complex numbers, you apply the distributive property, similar to multiplying algebraic expressions. When multiplying, it's important to remember the rule \(i^2 = -1\), which allows you to simplify your results.

• Consider two complex numbers: \((a + bi)(c + di)\). Their product is calculated as \((ac - bd) + (ad + bc)i\). Notice that the real part \(ac - bd\) incorporates the rule \(i^2 = -1\).
• As seen in the given exercise, we first express negative square roots using \(i\), transforming the multiplication. Then we multiply the transformed parts together.
• The property \(i\cdot i = i^2 = -1\) simplifies the expression further by converting the square of \(i\) into \(-1\). In the example \(\sqrt{-9} \cdot \sqrt{-6} = (i\sqrt{9})(i\sqrt{6}) = i^2\sqrt{54} = -\sqrt{54}\).

With these steps, you can effectively tackle multiplication involving complex numbers, performing accurate calculations and reaching simplified results.