Question

Find the value of \( (3 + \sqrt{8})^5 + (3 - \sqrt{8})^5 \):

• A. 6926
• B. 6826
• C. 6726
• D. 6626

Hint:

When dealing with sums of powers of conjugates, use the binomial expansion and simplify by considering the cancellation of odd and even powers of the terms.

Answer 1

The Correct Option is C

We are asked to find the value of \( (3 + \sqrt{8})^5 + (3 - \sqrt{8})^5 \). Step 1: Notice that this expression is a sum of the powers of two conjugates. We can use the binomial theorem to expand each term: \[ (3 + \sqrt{8})^5 = \sum_{k=0}^{5} \binom{5}{k} 3^{5-k} (\sqrt{8})^k \] \[ (3 - \sqrt{8})^5 = \sum_{k=0}^{5} \binom{5}{k} 3^{5-k} (-\sqrt{8})^k \] Step 2: Add the two expansions, and note that the odd powers of \( \sqrt{8} \) cancel out, leaving only the even powers. After simplifying the remaining terms, we get the value \( 6726 \). Thus, the correct answer is 6726.