Problem 44 - College Algebra

Question

Perform the indicated operations and write the result in standard form. $$ (3 \sqrt{-7})(2 \sqrt{-8}) $$

Step-by-Step Solution

Answer

The result of the multiplication \( (3 \sqrt{-7})(2 \sqrt{-8}) \) in standard form is \( -12 \sqrt{14} \)

Step 1: Write numbers in complex number form

Both numbers are written in the form of a product of a real number and the square root of a negative number. We can convert these to standard complex number representations by recognizing that the square root of a negative number is equal to \(i\) times the square root of the corresponding positive number. The two numbers are then converted to \( 3i \sqrt{7} \) and \( 2i \sqrt{8} \) respectively.

Step 2: Multiply the complex numbers

Next, we multiply the two complex numbers just like any other multiplication. \( (3i \sqrt{7})(2i \sqrt{8}) = 6i^2 \sqrt{56} \).

Step 3: Simplify the equation

Now we remember that \(i^2 = -1\) in complex numbers. Substituting this, \( 6i^2 \sqrt{56} = -6 \sqrt{56} \). The last step is to simplify \( \sqrt{56} \) which simplifies to \(2 \sqrt{14}\), so the final answer is \( -6*2\sqrt{14}=-12 \sqrt{14} \)

Key Concepts

Standard Form

When working with complex numbers, it's essential to know how to express them into their standard form. A complex number is typically written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The real part \( a \) stands for the magnitude along the horizontal axis, while \( bi \) denotes the magnitude along the imaginary axis.

• The standard form helps us to easily distinguish between the real and imaginary components of any complex number.
• When you are given numbers under the square root of a negative number, like \( \sqrt{-7} \) or \( \sqrt{-8} \), you transform them into a structure involving \( i \), such as \( i\sqrt{7} \) or \( i\sqrt{8} \).

This is crucial because it allows us to work with these numbers algebraically, making them easier to handle during operations like addition, subtraction, and especially multiplication. So, always convert to standard form when dealing with complex numbers.

Imaginary Unit

The imaginary unit, represented by \( i \), is a mathematical constant used to extend the real number system to complex numbers. The defining property of the imaginary unit is that \( i^2 = -1 \). This unique characteristic is what allows us to handle the square roots of negative numbers.

• When you encounter a square root of a negative number, such as \( \sqrt{-x} \), it translates to \( i\sqrt{x} \).
• The introduction of \( i \) simplifies calculations involving negative square roots, allowing for direct manipulation of complex numbers.
• In expressions involving powers of \( i \), you can use the cycle: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). This pattern repeats every four powers.

Understanding and working with \( i \) is fundamental when dealing with operations involving complex numbers, as it allows simple expression of what would otherwise be undefined within the reals.

Multiplying Complex Numbers

Multiplying complex numbers involves combining both the real and imaginary components using standard multiplication rules. Let’s go through what happens step by step:

• First, when multiplying two complex numbers like \( (a + bi) \) and \( (c + di) \), use the distributive law: \((ac + adi + bci + bdi^2)\).
• Recall that \( i^2 = -1 \), which simplifies the term \( bdi^2 \) to \(-bd\).
• The entire expression simplifies to \((ac - bd) + (ad + bc)i\).

In the problem involving \((3i \sqrt{7})(2i \sqrt{8})\), we first multiply the coefficients and then the \( i \) terms, resulting in \( i^2 \), which simplifies using \( i^2 = -1 \).
This step provides \(-6\sqrt{56} \), requiring further simplification of \( \sqrt{56} \) to \( 2\sqrt{14} \). Thus, the multiplication of two complex forms doesn't just multiply the components but also results in translation from something seemingly complex into a simpler, understandable real or complex number.