- Write \(0.98\) as \((1 + (-0.02))\), so \(a=1\), \(b=-0.02\), and \(n=5\).
- Apply the Binomial Theorem: \((1 - 0.02)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-0.02)^k\).
- Calculate the terms: \(1, -0.1, 0.004, -0.00008, 0.0000008, -0.000000032\).
- Sum the results to obtain \(\boxed{0.903920768}\).
Explanation
- Identifying Variables for the Binomial Theorem
To evaluate \(0.98^5\) using the Binomial Theorem, we first need to express \(0.98\) as a binomial of the form \((a + b)^n\). The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
To fit the form \((1 + \Box)^n\), we set \(a = 1\). Then we find \(b\) such that \(a + b = 0.98\), which means \(1 + b = 0.98\), giving \(b = -0.02\). The exponent is \(n = 5\).
Thus, we have:
\(a = 1\)
\(b = -0.02\)
\(n = 5\)
Our expression is \((1 + (-0.02))^5\).
- Expanding the Binomial Expression
Now we apply the Binomial Theorem:
\((1 - 0.02)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-0.02)^k\)
Expanding this:
\(= \binom{5}{0}(1)^5(-0.02)^0 + \binom{5}{1}(1)^4(-0.02)^1 + \binom{5}{2}(1)^3(-0.02)^2 + \binom{5}{3}(1)^2(-0.02)^3 + \binom{5}{4}(1)^1(-0.02)^4 + \binom{5}{5}(1)^0(-0.02)^5\)
Calculating the coefficients \(\binom{5}{k}\):
\(\binom{5}{0} = 1\), \(\binom{5}{1} = 5\), \(\binom{5}{2} = 10\), \(\binom{5}{3} = 10\), \(\binom{5}{4} = 5\), \(\binom{5}{5} = 1\)
Now calculating each term:
Term 0: \(1 \times 1 \times 1 = 1\)
Term 1: \(5 \times 1 \times (-0.02) = -0.1\)
Term 2: \(10 \times 1 \times 0.0004 = 0.004\)
Term 3: \(10 \times 1 \times (-0.000008) = -0.00008\)
Term 4: \(5 \times 1 \times 0.00000016 = 0.0000008\)
Term 5: \(1 \times 1 \times (-0.000000032) = -0.000000032\)
- Final Calculation
Finally, we sum all the terms:
\(1 - 0.1 + 0.004 - 0.00008 + 0.0000008 - 0.000000032\)
Summing these up step by step:
\(1 - 0.1 = 0.9\)
\(0.9 + 0.004 = 0.904\)
\(0.904 - 0.00008 = 0.90392\)
\(0.90392 + 0.0000008 = 0.9039208\)
\(0.9039208 - 0.000000032 = 0.903920768\)
Thus, the value of \(0.98^5\) is \(\boxed{0.903920768}\).
Examples
The Binomial Theorem is a powerful tool for quick approximations in finance and engineering. For example, if you are calculating compound interest over a very small period or estimating depreciation, instead of long multiplication, you can use the first few terms of the binomial expansion to get a very accurate result quickly. This is essential for back-of-the-envelope calculations when precision is needed without full computational power.