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Multiply and simplify (6 + 8i)(6 - 8i) and choose the correct option.

Mathematics

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Answer

A

Step-by-step solution

4 steps

To multiply and simplify (6+8i)(6−8i)(6 + 8i)(6 - 8i), we can use the difference of squares formula, which states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. In the case of complex numbers, where i2=−1i^2 = -1, the product of complex conjugates (a+bi)(a−bi)(a+bi)(a-bi) simplifies to a2+b2a^2 + b^2.

  1. Identify aa and bb from the expression.

    In (6+8i)(6−8i)(6 + 8i)(6 - 8i), we have a=6a = 6 and b=8b = 8.

  2. Apply the formula for the product of complex conjugates.

    The formula is a2+b2a^2 + b^2.

  3. Substitute the values of aa and bb into the formula.

    62+826^2 + 8^2

  4. Calculate the squares and sum them.

    36+6436 + 64 100100

    The simplified product is 100.

    Comparing this to the given options:
    A. 100
    B. 80
    C. 60
    D. 40

    The correct option is A.

    The final answer is A. 100\boxed{\text{A. 100}}.

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