Find the value of `(98)^(4)` by using the binomial theorem.

Options

A. 92236846
B. 92239816
C. 9233886
D. 92236806

Explanation

To find the value of \( (98)^4 \) using the binomial theorem, we can express \( 98 \) as \( 100 - 2 \). Thus, we rewrite the expression as:

\[ (98)^4 = (100 - 2)^4 \]

Now, we can apply the binomial theorem, which states that:

\[ (a - b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} (-b)^r \]

In our case, \( a = 100 \), \( b = 2 \), and \( n = 4 \). Therefore, we can expand \( (100 - 2)^4 \) as follows:

\[ (100 - 2)^4 = \sum_{r=0}^{4} \binom{4}{r} (100)^{4-r} (-2)^r \]

Now, we will calculate each term in the expansion:

1. **For \( r = 0 \)**:
\[ \binom{4}{0} (100)^{4} (-2)^{0} = 1 \cdot 100^4 \cdot 1 = 100000000 \]

2. **For \( r = 1 \)**:
\[ \binom{4}{1} (100)^{3} (-2)^{1} = 4 \cdot 100^3 \cdot (-2) = -8000000 \]

3. **For \( r = 2 \)**:
\[ \binom{4}{2} (100)^{2} (-2)^{2} = 6 \cdot 100^2 \cdot 4 = 2400000 \]

4. **For \( r = 3 \)**:
\[ \binom{4}{3} (100)^{1} (-2)^{3} = 4 \cdot 100 \cdot (-8) = -3200 \]

5. **For \( r = 4 \)**:
\[ \binom{4}{4} (100)^{0} (-2)^{4} = 1 \cdot 1 \cdot 16 = 16 \]

Now, we can sum all these terms together:

\[ 100000000 - 8000000 + 2400000 - 3200 + 16 \]

Calculating this step by step:

- First, \( 100000000 - 8000000 = 92000000 \)
- Next, \( 92000000 + 2400000 = 94400000 \)
- Then, \( 94400000 - 3200 = 94396800 \)
- Finally, \( 94396800 + 16 = 94396816 \)

Thus, the value of \( (98)^4 \) is:

\[ \boxed{94396816} \]