Using Binomial theorem, evaluate (99)^5.

To evaluate \( (99)^5 \) using the Binomial theorem, we rewrite \( 99 \) as \( (100 - 1) \), and then expand \( (100 - 1)^5 \). Here's the step-by-step solution:

### Step 1: Write the expression using binomial expansion
The Binomial theorem states:
\[
(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
\]
Here, \( a = 100 \), \( b = -1 \), and \( n = 5 \). Therefore:
\[
(99)^5 = (100 - 1)^5 = \sum_{k=0}^5 \binom{5}{k} (100)^{5-k} (-1)^k
\]

### Step 2: Expand the terms
We calculate each term of the expansion for \( k = 0 \) to \( k = 5 \). The general term is:
\[
T_k = \binom{5}{k} (100)^{5-k} (-1)^k
\]

#### For \( k = 0 \):
\[
T_0 = \binom{5}{0} (100)^5 (-1)^0 = 1 \cdot 100^5 \cdot 1 = 100^5
\]

#### For \( k = 1 \):
\[
T_1 = \binom{5}{1} (100)^4 (-1)^1 = 5 \cdot 100^4 \cdot (-1) = -5 \cdot 100^4
\]

#### For \( k = 2 \):
\[
T_2 = \binom{5}{2} (100)^3 (-1)^2 = 10 \cdot 100^3 \cdot 1 = 10 \cdot 100^3
\]

#### For \( k = 3 \):
\[
T_3 = \binom{5}{3} (100)^2 (-1)^3 = 10 \cdot 100^2 \cdot (-1) = -10 \cdot 100^2
\]

#### For \( k = 4 \):
\[
T_4 = \binom{5}{4} (100)^1 (-1)^4 = 5 \cdot 100^1 \cdot 1 = 5 \cdot 100
\]

#### For \( k = 5 \):
\[
T_5 = \binom{5}{5} (100)^0 (-1)^5 = 1 \cdot 1 \cdot (-1) = -1
\]

### Step 3: Write the full expansion
Adding all the terms:
\[
(99)^5 = T_0 + T_1 + T_2 + T_3 + T_4 + T_5
\]
\[
(99)^5 = 100^5 - 5 \cdot 100^4 + 10 \cdot 100^3 - 10 \cdot 100^2 + 5 \cdot 100 - 1
\]

### Step 4: Simplify each power of 100
1. \( 100^5 = 10^{10} = 10000000000 \)
2. \( 100^4 = 10^8 = 100000000 \)
3. \( 100^3 = 10^6 = 1000000 \)
4. \( 100^2 = 10^4 = 10000 \)
5. \( 100 = 10^2 = 100 \)

Now substitute these values into the expression:
\[
(99)^5 = 10000000000 - 5 \cdot 100000000 + 10 \cdot 1000000 - 10 \cdot 10000 + 5 \cdot 100 - 1
\]

### Step 5: Perform the arithmetic
1. \( 5 \cdot 100000000 = 500000000 \)
2. \( 10 \cdot 1000000 = 10000000 \)
3. \( 10 \cdot 10000 = 100000 \)
4. \( 5 \cdot 100 = 500 \)

Substitute these into the equation:
\[
(99)^5 = 10000000000 - 500000000 + 10000000 - 100000 + 500 - 1
\]

Now simplify step-by-step:
\[
(99)^5 = 10000000000 - 500000000 = 9500000000
\]
\[
9500000000 + 10000000 = 9510000000
\]
\[
9510000000 - 100000 = 9509900000
\]
\[
9509900000 + 500 = 9509900500
\]
\[
9509900500 - 1 = 9509900499
\]

### Final Answer:
The value of \( (99)^5 \) is **9509900499**.