Simplify the following without a calculator: \( \left(\sqrt{7}+\frac{1}{\sqrt{7}}\right)\left(2 \sqrt{7}-\frac{1}{\sqrt{7}}\right)\left(\frac{1}{\sqrt{7}}\right) \)

To simplify the expression \( \left(\sqrt{7}+\frac{1}{\sqrt{7}}\right)\left(2 \sqrt{7}-\frac{1}{\sqrt{7}}\right)\left(\frac{1}{\sqrt{7}}\right) \), we can first focus on simplifying the product of the first two terms: Let \( a = \sqrt{7} \) so that the expression becomes: \[ \left(a + \frac{1}{a}\right)\left(2a - \frac{1}{a}\right)\left(\frac{1}{a}\right) \] We start by simplifying \( (a + \frac{1}{a})(2a - \frac{1}{a}) \): \[ = a(2a) + a\left(-\frac{1}{a}\right) + \frac{1}{a}(2a) + \frac{1}{a}\left(-\frac{1}{a}\right) \] \[ = 2a^2 - 1 + 2 - \frac{1}{a^2} \] \[ = 2a^2 - 1 + 2 - \frac{1}{a^2} \] \[ = 2a^2 + 1 - \frac{1}{a^2} \] Replacing \( a^2 \) with \( 7 \): \[ = 2(7) + 1 - \frac{1}{7} \] \[ = 14 + 1 - \frac{1}{7} \] \[ = 15 - \frac{1}{7} \] \[ = \frac{105}{7} - \frac{1}{7} = \frac{104}{7} \] Now, we have the result of the product of the first two terms, which we can multiply by \( \frac{1}{a} \) (or \( \frac{1}{\sqrt{7}} \)): \[ \frac{104}{7} \cdot \frac{1}{\sqrt{7}} = \frac{104}{7\sqrt{7}} \] To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{7} \): \[ = \frac{104\sqrt{7}}{7 \cdot 7} = \frac{104\sqrt{7}}{49} \] So the final simplified expression is: \[ \frac{104\sqrt{7}}{49} \]