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Math  /  Algebra

Question2) If ii represents the imaginary unit, which of the following is yi(y−4i)2y i(y-4 i)^{2} in simplest form? (1) y2i+4yy^{2} i+4 y (2) 8y2+y3i−16yi8 y^{2}+y^{3} i-16 y i (3) y3i−8y2−16yiy^{3} i-8 y^{2}-16 y i (4) y3i−8y2i2−16yiy^{3} i-8 y^{2} i^{2}-16 y i

Studdy Solution

STEP 1

1. i i is the imaginary unit, where i2=−1 i^2 = -1 .
2. We need to simplify the expression yi(y−4i)2 y i (y - 4i)^2 .
3. The expression can be expanded and simplified using algebraic identities.

STEP 2

1. Expand the expression (y−4i)2 (y - 4i)^2 .
2. Multiply the result by yi y i .
3. Simplify the expression using the property i2=−1 i^2 = -1 .
4. Compare the simplified expression with the given options.

STEP 3

Expand the expression (y−4i)2 (y - 4i)^2 using the formula for the square of a binomial:
(y−4i)2=y2−2⋅y⋅4i+(4i)2 (y - 4i)^2 = y^2 - 2 \cdot y \cdot 4i + (4i)^2
=y2−8yi+16i2 = y^2 - 8yi + 16i^2
Since i2=−1 i^2 = -1 , substitute:
=y2−8yi+16(−1) = y^2 - 8yi + 16(-1)
=y2−8yi−16 = y^2 - 8yi - 16

STEP 4

Multiply the expanded expression by yi y i :
yi(y2−8yi−16) y i (y^2 - 8yi - 16)
Distribute yi y i across each term:
=yi⋅y2−yi⋅8yi−yi⋅16 = y i \cdot y^2 - y i \cdot 8yi - y i \cdot 16
=y3i−8y2i2−16yi = y^3 i - 8y^2 i^2 - 16yi

STEP 5

Simplify the expression using i2=−1 i^2 = -1 :
=y3i−8y2(−1)−16yi = y^3 i - 8y^2(-1) - 16yi
=y3i+8y2−16yi = y^3 i + 8y^2 - 16yi

STEP 6

Compare the simplified expression y3i+8y2−16yi y^3 i + 8y^2 - 16yi with the given options:
The expression matches option (3):
y3i−8y2−16yi y^3 i - 8 y^2 - 16 y i
Thus, the simplest form of the expression is:
y3i−8y2−16yi \boxed{y^3 i - 8 y^2 - 16 y i}

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