Question

Evaluate the expression for the given values of the variables. $$ (a-b)^{4} \text { when } a=4 \text { and } b=2 $$

Short answer

The result of the expression \((a-b)^4\) when \(a=4\) and \(b=2\) is 16.

Step-by-step solution

Understand the problem

The exercise is asking to evaluate the expression \((a-b)^4\) with given values \(a=4\) and \(b=2\).

Expand the Binomial

Use binomial theorem to expand the expression. The binomial theorem states \((a-b)^n = a^n - na^{n-1}b + \binom{n}{2}a^{n-2}b^2 - \binom{n}{3}a^{n-3}b^3 + \ldots + (-1)^nb^n\). Using this theorem, \((a-b)^4\) expands to \(a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4\).

Substitute the given values

Now substitute the given values into the expanded equation, i.e. replace \(a\) with 4 and \(b\) with 2. This results in \(4^4 - 4*4^3*2 + 6*4^2*2^2 - 4*4*2^3 + 2^4\).

Compute the solution

After substitution, simplify the arithmetic expression to find the answer. After simplifying the expression, the result is 16.