Simplify and write in standard form.
1.)
(radical 3 + i radical 15)(radical 3 - i radical 15)
2.)
8 + 16i / 2i
Thanks again everyone. Any help is appreciated.
1.
(√3 + i√15)(√3 - i√15)
= 3 - 15i^2
= 3 - 15(-1) = 18
2. I am pretty sure you meant:
(8+16i)(2i)
= (8 + 16i)/(2i) * i/i
= (8i + 16i^2)/(2i^2)
= (8i - 16)/-2
= 8 - 4i
To simplify and write the given expressions in standard form, we can follow the steps below for each question:
1.) Simplifying and writing in standard form:
To simplify the expression (radical 3 + i radical 15)(radical 3 - i radical 15), we will use the difference of squares formula, which states that (a - b)(a + b) = a^2 - b^2. In this case, a = radical 3 and b = i radical 15.
(a) Calculate the squares of a and b:
a^2 = (radical 3)^2 = 3
b^2 = (i radical 15)^2 = -15
(b) Use the difference of squares formula:
(radical 3 + i radical 15)(radical 3 - i radical 15) = (radical 3)^2 - (i radical 15)^2
Substitute the calculated values:
= 3 - (-15)
= 3 + 15
= 18
Therefore, (radical 3 + i radical 15)(radical 3 - i radical 15) simplifies to 18.
2.) Simplifying and writing in standard form:
To simplify the expression (8 + 16i) / 2i, we need to divide the numerator by the denominator.
(a) Divide the real parts:
8 / 2 = 4
(b) Divide the imaginary parts:
16i / 2i = 8i / i
To simplify further, we can multiply by the conjugate of the denominator, which is -i.
8i / i = (8i * -i) / (i * -i)
= -8i^2 / -1 (using the fact that i^2 = -1)
= -8(-1) / -1
= 8
Therefore, (8 + 16i) / 2i simplifies to 8.
In standard form, the answers are:
1.) 18
2.) 8