Which of the following is equivalent to the expression (i⋅5–√)2⋅3?
explain how you would get the answer
To simplify the expression (i⋅5–√)2⋅3, we first need to determine the value of the term inside the parentheses, i⋅5–√.
Let's break it down step by step:
1. Start with the term i⋅5.
- We can multiply the imaginary unit i (which represents √(-1)) by 5.
- The product is 5i.
2. Add the square root (√) of the result from step 1 to 5i.
- We take the square root of 5i, which is √(5i).
- The expression i⋅ 5 – √ becomes 5i + √(5i).
Now, we substitute this simplified expression back into the original expression provided:
(5i + √(5i))2⋅3
To simplify further, we square the term inside the parentheses:
= (5i + √(5i))2
= (5i + √(5i))(5i + √(5i))
= 25i2 + 5i√(5i) + 5i√(5i) + (√(5i))2
= 25(-1) + 10i√(5i) + (√(5i))2
= -25 + 10i√(5i) + 5i
= -25 + 10i√(5i) + 5i
= -25 + 15i + 10i√(5i)
Finally, we multiply this entire expression by 3:
-25(3) + 15i(3) + 10i√(5i)(3)
= -75 + 45i + 30i√(5i)
So, the equivalent expression is -75 + 45i + 30i√(5i).