Problem 68 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$i(2+7 i)(2-7 i)$$

Step-by-Step Solution

Answer

The product in standard form is 0 + 53i.

Step 1: Identify the expression

We start by identifying the expression that needs to be simplified: \( i(2+7i)(2-7i) \). This is a complex number multiplication.

Step 2: Use the formula for the difference of squares

The expression \((2+7i)(2-7i)\) is the difference of squares. Recall that \((a+b)(a-b) = a^2 - b^2\). Here, \( a = 2 \) and \( b = 7i \).

Step 3: Calculate the squares

Calculate \(a^2 = 2^2 = 4\) and \(b^2 = (7i)^2 = 49i^2\). Remember that \(i^2 = -1\), so \(b^2 = 49(-1) = -49\).

Step 4: Apply the difference of squares

Substitute the squares back into the formula: \((2+7i)(2-7i) = 4 - (-49) = 4 + 49 = 53\).

Step 5: Multiply by the remaining factor

Multiply the result by \(i\): \( i \times 53 = 53i \).

Step 6: Write the product in standard form

Since the standard form of a complex number is \(a + bi\), the product \(53i\) can be written as \(0 + 53i\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity used to simplify expressions. It states that for any two numbers, \(a\) and \(b\), the expression \((a+b)(a-b)\) can be simplified to \(a^2 - b^2\). This identity is particularly useful when dealing with complex numbers.

In the original exercise, we apply the difference of squares to the expression \((2+7i)(2-7i)\). Here, \(a = 2\) and \(b = 7i\), making the expression a perfect candidate for this identity.

Let's break it down further:

• Calculate \(a^2\): \(a = 2\) so \(a^2 = 4\).
• Calculate \(b^2\): \(b = 7i\) so \((7i)^2 = 49i^2\). Since \(i^2 = -1\), this becomes \(49(-1) = -49\).
• Apply the difference: Substitute back to get \(a^2 - b^2 = 4 - (-49) = 4 + 49\).

This gives us \(53\) as the result of the difference of squares, thus simplifying the original complex multiplication.

Standard Form

The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. This helps clearly distinguish the real component, \(a\), from the imaginary component, \(b\). Complex numbers are crucial in various fields of science and engineering, making understanding their structure significant.

In the exercise, once the multiplication is carried out and simplified using the difference of squares, we arrive at \(53i\). Our task is to convert this result into standard form.

• Since there is no real part in \(53i\), we denote the real part as zero, therefore expressing the number as \(0 + 53i\).

Remember:

• "\(a\)" is the real part. In this example, it is zero.
• "\(bi\)" is the imaginary part. Here, \(b = 53\).
  

This form allows one to easily distinguish and calculate with both components, efficiently utilizing complex numbers in equations and functions.

Complex Number Multiplication

Multiplying complex numbers involves both multiplication and the application of principles from algebra. When multiplying, each term of the first complex number is multiplied by each term of the second. Careful attention is given to imaginary components because of the property \(i^2 = -1\).

In the given exercise \(i(2+7i)(2-7i)\), multiplication is tackled in steps:

• First, tackle \((2+7i)(2-7i)\). Use the difference of squares method, yielding \(53\).
• Next, multiply by the remaining factor: \(i \times 53\).
  

To multiply a complex number by \(i\), each component is affected:

• The real part becomes the imaginary part: for example, if its result were \(53 + 0i\), then it would switch to \(0 + 53i\).
• The imaginary part is already solely dependent on \(i\), hence remains in the result as part of the product.

This process exemplifies how multiplication shapes the final form of complex numbers, especially when expressed in standard form, demonstrating their versatility in both real and imaginary number interactions.