Question

Find the value of \((98)^{4}\) by using the binomial theorem. (1) 92236846 (2) 92236816 (3) 92236886 (4) 92236806

Short answer

Answer: 92236816

Step-by-step solution

Write the term in binomial form

We need to write 98 in binomial form. We can write 98 as \((100 - 2)\). So, the given exercise becomes finding the value of \((100 - 2)^{4}\).

Apply the binomial theorem

We will apply the binomial theorem on \((100 - 2)^{4}\). The binomial theorem states that for any integers \(a\), \(b\), and \(n\): \((a + b)^{n} = \displaystyle\sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\) In our case, \(a = 100\), \(b = -2\), and \(n = 4\). Applying the binomial theorem, \((100 - 2)^{4} = \displaystyle\sum_{k=0}^{4} \binom{4}{k}(100)^{4-k}(-2)^k\).

Calculate the terms of the binomial expansion

We need to find the terms of this binomial expansion. There will be 5 terms for this expansion since \(n = 4\). The terms are calculated as follows: \(\binom{4}{0}(100)^{4}(-2)^0 = 1(1000000)(1) = 1000000\) \(\binom{4}{1}(100)^{3}(-2)^1 = 4(100000)(-2) = -800000\) \(\binom{4}{2}(100)^{2}(-2)^2 = 6(10000)(4) = 240000\) \(\binom{4}{3}(100)^{1}(-2)^3 = 4(100)(-8) = -3200\) \(\binom{4}{4}(100)^{0}(-2)^4 = 1(1)(16) = 16\)

Sum the terms of the expansion

Now, we will find the sum of these terms: \((100 - 2)^{4} = 1000000 - 800000 + 240000 - 3200 + 16\) \((100 - 2)^{4} = 92236816\) So, the value of \((98)^{4}\) is 92236816, which corresponds to option (2).

Key concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers. The first number is called the base and the second, the exponent. This process involves multiplying the base by itself for the number of times indicated by the exponent. For example, in \((98)^{4}\), 98 is the base, and 4 is the exponent.

Here's a simple way to understand it:

• If the exponent is 1, \(a^1 = a\).
• If the exponent is 2, \(a^2 = a \times a\).
• If the exponent is 3, \(a^3 = a \times a \times a\), and so on.

Using this principle, to calculate \((98)^4\), you would normally multiply: \(98 \times 98 \times 98 \times 98\). However, this can be cumbersome, which is why techniques like polynomial expansion are helpful here.

Polynomial Expansion

Polynomial expansion involves expressing a polynomial that is raised to a power, like \((a + b)^n\), as a sum of terms using coefficients. The binomial theorem is a common formula used for such expansions and states:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\]This formula breaks down the complex expression \((a + b)^n\) into manageable parts. Here's how it works:

- Each term in the expansion contains the base \(a\) and the other number \(b\) raised to different powers that sum up to \(n\).- The binomial coefficient \(\binom{n}{k}\) determines how many times a specific term appears in the expansion.

Consider the example \((100 - 2)^4\) in the exercise. From the binomial expansion, we get several terms. Each consists of a binomial coefficient, followed by powers of \(100\) and \(-2\) such that their powers add up to 4. This method simplifies multiplication and provides a systematic way to expand powers of binomials.

Combinatorics

Combinatorics is an area of mathematics that deals with counting and arranging possibilities. In the context of the binomial theorem, it provides us with the binomial coefficients \(\binom{n}{k}\). The coefficients in binomial expansions are directly linked to combinations, where \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements:

\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

Here, "!" denotes factorial, which means multiplying a series of descending positive integers.

• For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Understanding these coefficients plays a crucial role in problems involving polynomial expansions like our exercise, since they determine the weight of each term in the final expanded form. In the calculation of \((100-2)^4\), each term's coefficient was found using combinations, making this mathematical concept an essential component.