Write the Expression as a complex number in standard form.
5i(3+2i)(8+3i)
just expand as you normally would a polynomial, and you get
30i^3 + 125i^2 + 120i
But, once you recall that
i^2 = -1
i^3 = -i, you have
-125 + 90i
To express the given expression as a complex number in standard form, we need to simplify the expression by performing the multiplication and combining like terms.
Let's start by simplifying the multiplication step by step:
1. Multiply the second and third terms using the distributive property:
(2i)(8+3i) = 16i + 6i^2
Note: i^2 is equal to -1 because i represents the imaginary unit.
2. Multiply the first term with the result from step 1:
5i(16i + 6i^2) = 5i(16i) + 5i(6i^2)
3. Simplify the above expression:
5i(16i) = 80i^2 = 80(-1) = -80
5i(6i^2) = 30i^3 = 30(-i) = -30i
4. Combine the results from steps 2 and 3:
5i(16i) + 5i(6i^2) = -80 - 30i
Now, we need to multiply the result from step 4 with the first term, 3+2i:
(3+2i)(-80 - 30i) = -80(3+2i) - 30i(3+2i)
Simplify the above expression:
-80(3+2i) = -240 - 160i
-30i(3+2i) = -90i - 60i^2 = -90i - 60(-1) = -90i + 60
Finally, combine the two terms:
-240 - 160i - 90i + 60
= -240 + 60 - 160i - 90i
= -180 - 250i
Therefore, the expression 5i(3+2i)(8+3i) can be written as a complex number in standard form as -180 - 250i.