Question

multiply, and then simplify if possible.
$(sqrt{6} - sqrt{7})^2$
$(sqrt{6} - sqrt{7})^2 = \square$
(simplify your answer. do not factor. type an exact answer, using rad

Explanation:

Step1: Apply the square formula

We use the formula \((a - b)^2 = a^2 - 2ab + b^2\), where \(a=\sqrt{6}\) and \(b = \sqrt{7}\). So \((\sqrt{6}-\sqrt{7})^2=(\sqrt{6})^2-2\times\sqrt{6}\times\sqrt{7}+(\sqrt{7})^2\).

Step2: Simplify each term

Simplify \((\sqrt{6})^2 = 6\), \((\sqrt{7})^2=7\), and \(2\times\sqrt{6}\times\sqrt{7}=2\sqrt{42}\). Then substitute these values back into the expression: \(6 - 2\sqrt{42}+7\).

Step3: Combine like terms

Combine the constant terms \(6\) and \(7\): \(6 + 7-2\sqrt{42}=13 - 2\sqrt{42}\).

Answer:

\(13 - 2\sqrt{42}\)