Problem 139 - College Algebra

Question

Find the exact value of \(\sqrt{13+\sqrt{2}+\frac{7}{3+\sqrt{2}}}\) without the use of a calculator.

Step-by-Step Solution

Answer

4

Step 1: Simplify the fraction

To simplify the fraction \(\frac{7}{3+\sqrt{2}}\), multiply the numerator and denominator by the conjugate of the denominator. That is, multiply by \(3-\sqrt{2}\). This gives \(\frac{7(3-\sqrt{2})}{3^2-(\sqrt{2})^2} = \frac{21-7\sqrt{2}}{7} = 3 - \sqrt{2}\). The fraction simplifies to \(3 - \sqrt{2}\)

Step 2: Substitute into the expression

Now substitute the simplified fraction back into the original expression, which gives \(\sqrt{13 + \sqrt{2} + (3 - \sqrt{2})}\). This can be further simplified to \(\sqrt{16} = 4\).

Step 3: Final simplification

The final expression has been simplified to 4, a much simpler number than the original expression.