Perform the indicated operations and write the result in standard form. (7-i)(2+3i)

Verified step by step guidance

1. Identify the problem as multiplying two complex numbers: \((7 - i)\) and \((2 + 3i)\).
2. Use the distributive property (FOIL method) to expand the product: multiply each term in the first complex number by each term in the second complex number.
3. Calculate each product: \(7 \times 2\), \(7 \times 3i\), \(-i \times 2\), and \(-i \times 3i\).
4. Combine the real parts and the imaginary parts separately. Remember that \(i^2 = -1\), so simplify \(-i \times 3i\) accordingly.
5. Write the final expression in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form means writing the result explicitly as a sum of a real number and an imaginary number, making it easier to interpret and use in further calculations.

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method) to expand the product. Multiply each term in the first complex number by each term in the second, then combine like terms, remembering that i² = -1.

Simplifying Using i² = -1

Since i is the imaginary unit with the property i² = -1, any occurrence of i² in the expression should be replaced by -1. This simplification converts imaginary squared terms into real numbers, allowing the expression to be written in standard form.