What is the product of (2+\sqrt{13}) and (2-\sqrt{13})?

Correct answer: ( -9 )

The direct answer is B, \\(-9\\). The two expressions are conjugates: \\(2+\\sqrt{13}\\) and \\(2-\\sqrt{13}\\). Use the difference-of-squares identity \\((a+b)(a-b)=a^2-b^2\\). Here, \\(a=2\\) and \\(b=\\sqrt{13}\\). Therefore, \\((2+\\sqrt{13})(2-\\sqrt{13})=2^2-(\\sqrt{13})^2=4-13=-9\\). Option A, 9, has the wrong sign because 4 is smaller than 13. Option B, -9, is correct. Option C, 17, would result from adding 4 and 13, but multiplication of conjugates requires subtraction. Option D, \\(4+\\sqrt{13}\\), is not the result of multiplying the two binomials and leaves an unnecessary radical. You can also check by expansion: \\(4-2\\sqrt{13}+2\\sqrt{13}-13=4-13=-9\\); the middle terms cancel. Memory cue: conjugates multiply to the square of the rational part minus the square of the radical part. The supplied answer is correct.