Find each product. Write answers in standard form. (√6+i)(√6-i)

Verified step by step guidance

1. Recognize that the expression is a product of two complex conjugates: \((\sqrt{6} + i)(\sqrt{6} - i)\).
2. Recall the formula for the product of conjugates: \((a + b)(a - b) = a^2 - b^2\).
3. Identify \(a = \sqrt{6}\) and \(b = i\) in the expression.
4. Calculate \(a^2 = (\sqrt{6})^2 = 6\) and \(b^2 = i^2\); remember that \(i^2 = -1\).
5. Substitute these values into the formula to get \$6 - (-1)\(, which simplifies to \)6 + 1$.

Key Concepts

Complex Conjugates

Complex conjugates are pairs of complex numbers where the imaginary parts have opposite signs, such as (a + bi) and (a - bi). Multiplying conjugates results in a real number because the imaginary parts cancel out, simplifying expressions involving complex numbers.

Multiplying Binomials

Multiplying binomials involves applying the distributive property (FOIL method) to combine each term in the first binomial with each term in the second. This process expands the product into a polynomial, which can then be simplified by combining like terms.

Standard Form of Complex Numbers

The standard form of a complex number is written as a + bi, where a is the real part and b is the coefficient of the imaginary part i. Writing answers in standard form means expressing the result clearly as a sum of a real number and an imaginary number.