Byusing binomial theorem evaluate (i) `(107)^(5)` , (ii) `(55)^(3)`

Answer

To evaluate \( (107)^5 \) and \( (55)^3 \) using the Binomial Theorem, we proceed as follows:

### Part (i): Evaluate \( (107)^5 \)

1. **Rewrite the expression**:
\[ 107 = 100 + 7 \]
So, we can express \( (107)^5 \) as:
\[ (100 + 7)^5 \]

2. **Apply the Binomial Theorem**:
The Binomial Theorem states that:
\[ (x + y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r \]
Here, \( x = 100 \), \( y = 7 \), and \( n = 5 \).

3. **Expand using the theorem**:
\[ (100 + 7)^5 = \sum_{r=0}^{5} \binom{5}{r} (100)^{5-r} (7)^r \]
This gives us:
\[ = \binom{5}{0} (100)^5 (7)^0 + \binom{5}{1} (100)^4 (7)^1 + \binom{5}{2} (100)^3 (7)^2 + \binom{5}{3} (100)^2 (7)^3 + \binom{5}{4} (100)^1 (7)^4 + \binom{5}{5} (100)^0 (7)^5 \]

4. **Calculate each term**:
- For \( r = 0 \):
\[ \binom{5}{0} (100)^5 (7)^0 = 1 \times 10000000000 \times 1 = 10000000000 \]
- For \( r = 1 \):
\[ \binom{5}{1} (100)^4 (7)^1 = 5 \times 100000000 \times 7 = 3500000000 \]
- For \( r = 2 \):
\[ \binom{5}{2} (100)^3 (7)^2 = 10 \times 1000000 \times 49 = 490000000 \]
- For \( r = 3 \):
\[ \binom{5}{3} (100)^2 (7)^3 = 10 \times 10000 \times 343 = 34300000 \]
- For \( r = 4 \):
\[ \binom{5}{4} (100)^1 (7)^4 = 5 \times 100 \times 2401 = 120050 \]
- For \( r = 5 \):
\[ \binom{5}{5} (100)^0 (7)^5 = 1 \times 1 \times 16807 = 16807 \]

5. **Sum all the terms**:
\[ 10000000000 + 3500000000 + 490000000 + 34300000 + 120050 + 16807 = 11859225007 \]

Thus, \( (107)^5 = 11859225007 \).

### Part (ii): Evaluate \( (55)^3 \)

1. **Rewrite the expression**:
\[ 55 = 50 + 5 \]
So, we can express \( (55)^3 \) as:
\[ (50 + 5)^3 \]

2. **Apply the Binomial Theorem**:
Using the same theorem:
\[ (50 + 5)^3 = \sum_{r=0}^{3} \binom{3}{r} (50)^{3-r} (5)^r \]

3. **Expand using the theorem**:
\[ = \binom{3}{0} (50)^3 (5)^0 + \binom{3}{1} (50)^2 (5)^1 + \binom{3}{2} (50)^1 (5)^2 + \binom{3}{3} (50)^0 (5)^3 \]

4. **Calculate each term**:
- For \( r = 0 \):
\[ \binom{3}{0} (50)^3 (5)^0 = 1 \times 125000 = 125000 \]
- For \( r = 1 \):
\[ \binom{3}{1} (50)^2 (5)^1 = 3 \times 2500 \times 5 = 37500 \]
- For \( r = 2 \):
\[ \binom{3}{2} (50)^1 (5)^2 = 3 \times 50 \times 25 = 3750 \]
- For \( r = 3 \):
\[ \binom{3}{3} (50)^0 (5)^3 = 1 \times 1 \times 125 = 125 \]

5. **Sum all the terms**:
\[ 125000 + 37500 + 3750 + 125 = 162375 \]

Thus, \( (55)^3 = 162375 \).

### Final Answers:
- \( (107)^5 = 11859225007 \)
- \( (55)^3 = 162375 \)