Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(\sqrt{14}+\sqrt{10} i)(\sqrt{14}-\sqrt{10} i)$$

Step-by-Step Solution

Answer

The result of the operation in standard form is 4.

Step 1: Apply the distributive law

Apply the distributive law of multiplication over addition for the complex numbers. This law states that for any real numbers a, b, and c, the equation a * (b + c) = a * b + a * c holds true: \[(\sqrt{14}+\sqrt{10} i)(\sqrt{14}-\sqrt{10} i) = \sqrt{14} * \sqrt{14} + \sqrt{14} * -\sqrt{10}i + \sqrt{10}i * \sqrt{14} + \sqrt{10}i * -\sqrt{10}i\]

Step 2: Simplify the result

Simplify each term individually. The product of \(\sqrt{14} * \sqrt{14}\) gives the real number 14. The product of \(\sqrt{14} * -\sqrt{10}i\) and \(\sqrt{10}i * \sqrt{14}\) cancel each other out. Lastly the product of \(\sqrt{10}i * -\sqrt{10}i\) gives the real number -10. Thus the equation becomes: 14 + -10 = 4

Step 3: Write the final answer

Put together the resulting terms. The result is a real number which is the sum of the simplified terms: \(4\)

Key Concepts

Distributive Law

The distributive law is a fundamental principle in mathematics that applies to operations involving addition and multiplication. This law states that when a number multiplies a sum, you can distribute the multiplication across the terms inside the sum. For example, in arithmetic, you can express the operation \(a \times (b + c)\) as \((a \times b) + (a \times c)\). The distributive property is particularly useful with complex numbers, like in the given exercise. We have an expression like \((\sqrt{14}+\sqrt{10} i)(\sqrt{14}-\sqrt{10} i)\). Here, the multiplicative and additive components are separated using distribution.
Applying the distributive law ensures each term from the first binomial is multiplied by each term in the second binomial. This results in four products which might simplify neatly when combined, especially because of conjugate pairs.

Standard Form

The standard form for a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In the problem we solved, after using the distributive law and simplifying, we ended up with a real number: \(4\). This is still a valid complex number in standard form, where the imaginary part \(b = 0\). But why is the standard form important?
First, it gives a clear way to write any complex number, showing both its real and imaginary parts explicitly. Second, it aids in performing arithmetic with complex numbers, like addition, subtraction, multiplication, and division. By breaking complex numbers into this form, it simplifies operations that would otherwise become unwieldy.

Simplification in Mathematics

Simplification is the process of reducing an expression to its simplest form. With complex numbers, simplification often involves combining like terms and eliminating imaginary or unnecessary components. In the exercise, simplification resulted from multiplying the respective terms and combining them efficiently.
For this specific problem, simplifying the products, such as \(\sqrt{14} \times \sqrt{14} = 14\) and \(\sqrt{10}i \times -\sqrt{10}i = -10\), brings clarity. Here, symmetry and cancellation of the imaginary terms led to a real answer.
Simplification helps in making expressions easier to work with, understand, and compare. It's a critical skill in mathematics that enables problem-solvers to focus on the core components of a problem and reach conclusions more efficiently.