Problem 46 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-7} \sqrt{-10}$$

Step-by-Step Solution

Answer

\(-\sqrt{70}\)

Step 1: Express each number with 'i'

Convert the square roots of negative numbers into complex form using \(i\), where \(i^2 = -1\). \[\sqrt{-7} = i\sqrt{7}, \quad \sqrt{-10} = i\sqrt{10}\]

Step 2: Multiply the expressions

Multiply the two complex expressions:\[(i \sqrt{7})(i \sqrt{10})\]

Step 3: Apply the properties of 'i'

Use the fact that \(i^2 = -1\) to simplify:\[i^2 \cdot \sqrt{7} \cdot \sqrt{10} = -1 \cdot \sqrt{70}\]

Step 4: Simplify the square root

Combine and simplify the square root term:\[- \sqrt{70}\]

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the letter \(i\), is a fundamental concept in complex numbers. It provides a solution to the equation \(x^2 = -1\), which isn't solvable with real numbers since there is no real number whose square is negative. That's where the imaginary unit comes into play. By definition, \(i^2 = -1\).

Imaginary numbers are expressed in terms of this unit, \(i\). For example:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These powers of \(i\) repeat every four exponents, creating a cycle.

In the context of square roots of negative numbers, \(i\) allows us to express these roots in a form that can be manipulated algebraically. For instance, the square root of a negative number, like \(\sqrt{-7}\), can be rewritten as \(i\sqrt{7}\). This helps in simplifying operations involving complex numbers.

Square Roots

Square roots are a mathematical operation that find a number which, when multiplied by itself, gives the original number. While this is straightforward for positive numbers, taking the square root of negative numbers introduces a need for complex numbers.

Traditionally, the square root of a negative number is not defined in the set of real numbers. To resolve this, we use the imaginary unit \(i\). Specifically, the square root of a negative number \(-x\) can be expressed as \(i\sqrt{x}\).

Consider \(\sqrt{-7}\). We express this as \(i\sqrt{7}\). By introducing \(i\), it becomes possible to handle otherwise undefined operations, permitting further simplification with the help of algebraic rules.

When simplifying expressions, separating the square root of a negative number into its real and imaginary parts is vital. Each component can then be managed independently in calculations, as demonstrated in the original exercise and steps.

Multiplication of Complex Numbers

The multiplication of complex numbers extends the rules of regular multiplication to accommodate imaginary units. When two complex numbers in the form \((a + bi)\) and \((c + di)\) are multiplied, the process is similar to multiplying two binomials, with the added step of simplifying any instances of \(i^2\) to \(-1\). This is due to the property \(i^2 = -1\).

To multiply expressions like \((i\sqrt{7})(i\sqrt{10})\), distribute each part of the first expression through the second:

• \(i\times i = i^2 = -1\)
• \(\sqrt{7}\times \sqrt{10} = \sqrt{70}\)

Combine these results to achieve \(-\sqrt{70}\).

Each step in the multiplication of complex numbers must consider the imaginary unit's effect, especially when powers of \(i\) are involved. Simplifying \(i^2\) to \(-1\) is crucial to merge imaginary components into their appropriate simplified forms, aiding in clearer and efficient calculations.