Evaluate 3.039⁴ using binomial expansion?

To evaluate 3.039⁴ using binomial expansion, we can use the formula:

(a + b)ⁿ = C(n, 0) * aⁿ * b⁰ + C(n, 1) * aⁿ⁻¹ * b¹ + C(n, 2) * aⁿ⁻² * b² + ... + C(n, n) * a⁰ * bⁿ

In this case, a = 3.039 and b = 0.039. We want to find (3.039 + 0.039)⁴.

Using the formula, we have:

(3.039 + 0.039)⁴ = C(4, 0) * 3.039⁴ * 0.039⁰ + C(4, 1) * 3.039³ * 0.039¹ + C(4, 2) * 3.039² * 0.039² + C(4, 3) * 3.039¹ * 0.039³ + C(4, 4) * 3.039⁰ * 0.039⁴

Calculating each term:

C(4, 0) = 1
C(4, 1) = 4
C(4, 2) = 6
C(4, 3) = 4
C(4, 4) = 1

(3.039 + 0.039)⁴ = 1 * 3.039⁴ * 0.039⁰ + 4 * 3.039³ * 0.039¹ + 6 * 3.039² * 0.039² + 4 * 3.039¹ * 0.039³ + 1 * 3.039⁰ * 0.039⁴

Simplifying each term:

1 * 3.039⁴ * 0.039⁰ = 3.039⁴
4 * 3.039³ * 0.039¹ = 4 * 3.039³ * 0.039
6 * 3.039² * 0.039² = 6 * 3.039² * 0.039²
4 * 3.039¹ * 0.039³ = 4 * 3.039¹ * 0.039³
1 * 3.039⁰ * 0.039⁴ = 0.039⁴

Calculating each term:

3.039⁴ ≈ 48.724
4 * 3.039³ * 0.039 ≈ 4 * 44.214 * 0.039 ≈ 6.926
6 * 3.039² * 0.039² ≈ 6 * 9.256 * 0.001521 ≈ 0.083
4 * 3.039¹ * 0.039³ ≈ 4 * 3.039 * 0.000059 = 0.00072
0.039⁴ ≈ 0.000001

Adding all the terms:

48.724 + 6.926 + 0.083 + 0.00072 + 0.000001 ≈ 56.733

Therefore, 3.039⁴ ≈ 56.733 when evaluated using binomial expansion.