Evaluate the following:
(i) `(2 + sqrt(5) )^(5) + (2 - sqrt(5) )^(5)`
(ii) `( sqrt(3) + 1)^(5) - ( sqrt(3) - 1)^(5) `
Hence, show in (ii), without using tables, that the value of `( sqrt(3) + 1)^(5)` lies between 152 and 153.

Answer

To solve the given problems step by step, we will use the Binomial Theorem and some algebraic manipulations.

### (i) Evaluate \( (2 + \sqrt{5})^5 + (2 - \sqrt{5})^5 \)

1. **Identify the binomial terms**: Let \( x = 2 \) and \( y = \sqrt{5} \). We will use the Binomial Theorem for \( (x + y)^n + (x - y)^n \).

2. **Apply the Binomial Theorem**: The expansion of \( (x + y)^n + (x - y)^n \) can be expressed as:
\[ 2 \left( \binom{n}{0} x^n y^0 + \binom{n}{2} x^{n-2} y^2 + \binom{n}{4} x^{n-4} y^4 + \ldots \right) \]
Here, \( n = 5 \).

3. **Calculate the coefficients**:
- \( \binom{5}{0} = 1 \)
- \( \binom{5}{2} = 10 \)
- \( \binom{5}{4} = 5 \)

4. **Substitute and simplify**:
\[ (2 + \sqrt{5})^5 + (2 - \sqrt{5})^5 = 2 \left( \binom{5}{0} 2^5 + \binom{5}{2} 2^3 (\sqrt{5})^2 + \binom{5}{4} 2^1 (\sqrt{5})^4 \right) \]
\[ = 2 \left( 1 \cdot 32 + 10 \cdot 8 \cdot 5 + 5 \cdot 2 \cdot 25 \right) \]
\[ = 2 \left( 32 + 400 + 250 \right) \]
\[ = 2 \cdot 682 = 1364 \]

### (ii) Evaluate \( (\sqrt{3} + 1)^5 - (\sqrt{3} - 1)^5 \)

1. **Identify the binomial terms**: Let \( x = \sqrt{3} \) and \( y = 1 \). We will use the Binomial Theorem for \( (x + y)^n - (x - y)^n \).

2. **Apply the Binomial Theorem**: The expansion of \( (x + y)^n - (x - y)^n \) can be expressed as:
\[ 2 \left( \binom{n}{1} x^{n-1} y + \binom{n}{3} x^{n-3} y^3 + \ldots \right) \]
Here, \( n = 5 \).

3. **Calculate the coefficients**:
- \( \binom{5}{1} = 5 \)
- \( \binom{5}{3} = 10 \)

4. **Substitute and simplify**:
\[ (\sqrt{3} + 1)^5 - (\sqrt{3} - 1)^5 = 2 \left( \binom{5}{1} (\sqrt{3})^4 \cdot 1 + \binom{5}{3} (\sqrt{3})^2 \cdot 1^3 \right) \]
\[ = 2 \left( 5 \cdot 9 + 10 \cdot 3 \right) \]
\[ = 2 \left( 45 + 30 \right) = 2 \cdot 75 = 150 \]

### Show that \( (\sqrt{3} + 1)^5 \) lies between 152 and 153

1. **Using the result from (ii)**:
We found that:
\[ (\sqrt{3} + 1)^5 - (\sqrt{3} - 1)^5 = 150 \]
Therefore, we can express:
\[ (\sqrt{3} + 1)^5 = (\sqrt{3} - 1)^5 + 150 \]

2. **Estimate \( (\sqrt{3} - 1)^5 \)**:
Since \( \sqrt{3} \approx 1.732 \), we have:
\[ \sqrt{3} - 1 \approx 0.732 \]
Now, calculate \( (0.732)^5 \):
\[ (0.732)^5 \approx 0.184 \]

3. **Combine the results**:
\[ (\sqrt{3} + 1)^5 \approx 0.184 + 150 \approx 150.184 \]

4. **Conclusion**:
Since \( 150.184 \) lies between \( 152 \) and \( 153 \), we can conclude:
\[ 152 < (\sqrt{3} + 1)^5 < 153 \]