Question

simplify.
\\((5i)^{2}\\)
write your answer in the form \\(a + bi\\).

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). For \((5i)^2\), we have \(a = 5\), \(b = i\), and \(n = 2\). So, \((5i)^2 = 5^2 \times i^2\).

Step2: Calculate \(5^2\) and \(i^2\)

We know that \(5^2 = 25\) and by the definition of the imaginary unit \(i\), \(i^2=- 1\). Substituting these values in, we get \(25\times(-1)\).

Step3: Simplify the product

\(25\times(-1)=- 25\). We can write this in the form \(a + bi\) as \(-25+0i\) since there is no imaginary part (the coefficient of \(i\) is \(0\)).

Answer:

\(-25 + 0i\) (or simply \(-25\) as the \(0i\) is often omitted but still in the correct form)