Question

Perform the indicated operation, and write each expression in the standard form \(a+\) bi. $$ \sqrt{(3+4 i)(4 i-3)} $$

Short answer

5i

Step-by-step solution

- Recognize the expression

Identify the expression that needs to be simplified: \ \ \ \ \ \[ \sqrt{ (3 + 4i) (4i - 3) } \]

- Distribute the terms inside the square root

Apply the distributive property to the expression inside the square root: \ \ \ \ \[ (3 + 4i)(4i - 3) \] This results in: \ \ \[ 3 \times 4i + 3 \times (-3) + 4i \times 4i + 4i \times (-3) \]

- Simplify each distributive term

Now, find the product for each term: \ \ \[ 3 \times 4i = 12i \] \ \[ 3 \times (-3) = -9 \] \ \[ 4i \times 4i = 16i^2 \] \ \[ 4i \times (-3) = -12i \]

- Combine like terms

Combine like terms and substitute \(i^2 = -1\): \ \ \[ 12i - 12i + (-9) + 16i^2 \] Simplify by combining like terms: \ \ \[ 0 + (-9) + 16(-1) = -9 - 16 = -25 \]

- Simplify the square root

Find the square root of \(-25\): \ \ \[ \sqrt{-25} = \sqrt{-1 \times 25} = 5i \]

- Write in standard form

The expression in standard form \(a + bi\): \ \ \[ 0 + 5i = 5i \]

Key concepts

complex multiplication

To multiply complex numbers, we use a method that involves the distributive property. Consider two complex numbers, \(a + bi\) and \(c + di\). By applying the distributive property, we get: \((a + bi)(c + di) = ac + adi + bci + bdi^2\). We do each multiplication step-by-step and use the fact that \(i^2 = -1\). Therefore, the expression simplifies to: \((ac - bd) + (ad + bc)i \).
This is the product of two complex numbers in standard form, \(a + bi\).

👉 For example, in the given exercise, we have the complex numbers \((3 + 4i)\) and \((4i - 3)\). When applied the distributive method, we get the products: \3 \cdot 4i, 3 \cdot (-3), 4i \cdot 4i, \text{and} 4i \cdot (-3)\.
This step-by-step multiplication is crucial in simplifying the expression within the problem.

distributive property

The distributive property is a fundamental algebraic property used in the multiplication of complex numbers. It states that for any three numbers \(a\), \(b\), and \(c\), the following is true: \(a(b + c) = ab + ac \).

In the context of complex numbers, it means distributing each term in the first complex number by every term in the second complex number. Using this rule, we can simplify expressions involving complex numbers.
🔍 For instance, with \((3 + 4i)(4i - 3)\), you apply the distributive property to each term, leading to: \[3 \cdot 4i + 3 \cdot (-3) + 4i \cdot 4i + 4i \cdot (-3) \]
This results in \[12i - 9 + 16i^2 - 12i \].
Each intermediate step follows the distributive property to simplify the expression.

standard form of complex numbers

The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. In this form, \(a\) is the real part, and \(bi\) is the imaginary part.

✨ When working with complex numbers, always aim to present the final answer in this form for clarity. For the given exercise, after performing the multiplication and combining like terms, we simplify the expression as much as possible and present it as \[0 + 5i\] or simply \[5i\].
This step ensures our complex number is in its standard form, making further computations clearer and more manageable.

square roots of negative numbers

The square root of a negative number involves the imaginary unit, \(i\), where \( i = \sqrt{-1} \). Whenever we encounter a negative number under the square root symbol, it can be simplified by expressing the negative part as \(i\).

For instance, \ \sqrt{-25} \ = \ \sqrt{-1 \cdot 25} \ = 5i.
This conversion is essential in the context of complex numbers, as it transforms an otherwise undefined operation into a manageable complex number.
🤔 In our exercise, after simplifying the expression to \-25\, the next step was to find the square root, resulting in \ \sqrt{-25} \ = 5i.
Recognizing and applying this rule is crucial for correctly computing and simplifying expressions involving imaginary numbers.