Question

The sum of all rational terms in the expansion of $ \left( 2 + \sqrt{3} \right)^8 $ is

• A. \(16923\)
• B. \(3763\)
• C. \(33845\)
• D. \(18817\)

Hint:

When expanding expressions of the form \( (a + \sqrt{b})^n \), rational terms are those in which the exponent of \( \sqrt{b} \) is even. Use symmetry by adding \( (a + \sqrt{b})^n \) and \( (a - \sqrt{b})^n \) to quickly eliminate irrational components and double the rational terms.

Answer 1

The Correct Option is D

Let \( S = (2 + \sqrt{3})^8 \). To find the sum of all rational terms in the binomial expansion, we use: \[ (2 + \sqrt{3})^8 + (2 - \sqrt{3})^8 \] This removes all irrational terms since they cancel out in the symmetric expansion. 
So the sum of rational terms is: \[ \frac{(2 + \sqrt{3})^8 + (2 - \sqrt{3})^8}{2} \] 
We can also directly select terms where the exponent of \( \sqrt{3} \) is even (to ensure the term is rational). 
From the binomial expansion: \[ = \binom{8}{0}(2)^8 + \binom{8}{2}(2)^6(\sqrt{3})^2 + \binom{8}{4}(2)^4(\sqrt{3})^4 + \binom{8}{6}(2)^2(\sqrt{3})^6 + \binom{8}{8}(\sqrt{3})^8 \] \[ = 2^8 + 28 \cdot 2^6 \cdot 3 + 70 \cdot 2^4 \cdot 9 + 28 \cdot 2^2 \cdot 27 + 1 \cdot 81 \] \[ = 256 + 5376 + 10080 + 3024 + 81 = 18817 \]

Answer 2

To find the sum of all rational terms in the expansion of \( (2 + \sqrt{3})^8 \), we need to use the Binomial Theorem. The Binomial Theorem states:

\((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

For the expression \( (2 + \sqrt{3})^8 \), let \( a = 2 \) and \( b = \sqrt{3} \). Then:

\(\left(2 + \sqrt{3}\right)^8 = \sum_{k=0}^{8} \binom{8}{k} \cdot 2^{8-k} \cdot (\sqrt{3})^k\)

A term in this expansion will be rational if the power of \(\sqrt{3}\) is even. Therefore, we need terms for which \(k\) is even.

Let's find these coefficients and terms step-by-step:

1. Term for \(k = 0\):
   • The corresponding term is:
   • \(\binom{8}{0} \cdot 2^{8-0} \cdot (\sqrt{3})^0 = 1 \cdot 256 \cdot 1 = 256\)
2. Term for \(k = 2\):
   • The corresponding term is:
   • \(\binom{8}{2} \cdot 2^{6} \cdot (\sqrt{3})^2 = 28 \cdot 64 \cdot 3 = 5376\)
3. Term for \(k = 4\):
   • The corresponding term is:
   • \(\binom{8}{4} \cdot 2^{4} \cdot (\sqrt{3})^4 = 70 \cdot 16 \cdot 9 = 10080\)
4. Term for \(k = 6\):
   • The corresponding term is:
   • \(\binom{8}{6} \cdot 2^{2} \cdot (\sqrt{3})^6 = 28 \cdot 4 \cdot 27 = 3024\)
5. Term for \(k = 8\):
   • The corresponding term is:
   • \(\binom{8}{8} \cdot 2^{0} \cdot (\sqrt{3})^8 = 1 \cdot 1 \cdot 81 = 81\)

Now we sum all the rational terms:

\(256 + 5376 + 10080 + 3024 + 81 = 18817\)

Thus, the sum of all rational terms in the expansion of \( (2 + \sqrt{3})^8 \) is \(18817\).