Question

Using Binomial Theorem, Evaluate: \( (102)^5 \)

Hint:

To evaluate powers of numbers close to a round number, use the binomial expansion to simplify the calculation.

Answer 1

Step 1: Express \( 102 \) as \( 100 + 2 \).
We will apply the binomial theorem to evaluate \( (102)^5 \). First, express \( 102 \) as \( 100 + 2 \): \[ (102)^5 = (100 + 2)^5 \]
Step 2: Apply the binomial theorem.
The binomial expansion of \( (a + b)^n \) is given by: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] For \( a = 100 \), \( b = 2 \), and \( n = 5 \), we apply the binomial theorem: \[ (100 + 2)^5 = \sum_{k=0}^{5} \binom{5}{k} 100^{5-k} 2^k \]
Step 3: Calculate the terms.
Now, calculate each term of the expansion: \[ \binom{5}{0} 100^5 2^0 = 1 \times 100^5 \times 1 = 10000000000 \] \[ \binom{5}{1} 100^4 2^1 = 5 \times 100^4 \times 2 = 1000000000 \] \[ \binom{5}{2} 100^3 2^2 = 10 \times 100^3 \times 4 = 40000000 \] \[ \binom{5}{3} 100^2 2^3 = 10 \times 100^2 \times 8 = 800000 \] \[ \binom{5}{4} 100^1 2^4 = 5 \times 100 \times 16 = 8000 \] \[ \binom{5}{5} 100^0 2^5 = 1 \times 1 \times 32 = 32 \]
Step 4: Add the terms.
Now, add all the terms together: \[ (102)^5 = 10000000000 + 1000000000 + 40000000 + 800000 + 8000 + 32 \] \[ (102)^5 = 10000000000 + 1000000000 + 40000000 + 800000 + 8000 + 32 = 10510180032 \]
Step 5: Conclusion.
Thus, the value of \( (102)^5 \) is: \[ \boxed{10510180032} \]