Simplify 2(3-i^14)- (5-i^23)
3-i
Simplify: (5-3i)^2
16+30i
Simplify (3+2i) / (4-5i)
(2/41)-(23/41)i
is this right?
the second one should be
(5-3i)^2
= 25 - 30i + 9i^2
= 16 - 30i
the other two are correct
To simplify complex number expressions, we need to use the rules of complex arithmetic. Let's go through each question to check if the answers are correct.
1. Simplify 2(3-i^14) - (5-i^23)
To simplify this expression, we need to first evaluate the powers of "i". Let's calculate them:
i^14 = i^(4*3 + 2) = (i^4)^3 * i^2 = 1^3 * i^2 = 1 * (-1) = -1
i^23 = i^(4*5 + 3) = (i^4)^5 * i^3 = 1^5 * i^3 = 1 * (-i) = -i
Now, substitute the evaluated powers back into the expression:
2(3 - (-1)) - (5 - (-i))
= 2(3 + 1) - (5 + i)
= 2 * 4 - 5 - i
= 8 - 5 - i
= 3 - i
So, the answer is 3 - i, which matches your result.
2. Simplify (5 - 3i)^2
To simplify this expression, we square both the real and imaginary parts of the complex number (5 - 3i). Let's calculate it:
(5 - 3i)^2
= 5^2 - 2 * 5 * 3i + (3i)^2
= 25 - 30i + 9i^2
= 25 - 30i + 9(-1)
= 25 - 30i - 9
= 16 - 30i
So, the answer is 16 - 30i, which is different from your result. It seems like you made a mistake in the calculation by forgetting to change the sign of 9i^2 to -9.
3. Simplify (3 + 2i) / (4 - 5i)
To simplify this expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. Let's calculate it:
(3 + 2i) / (4 - 5i) * (4 + 5i) / (4 + 5i)
= [(3 + 2i)(4 + 5i)] / [(4 - 5i)(4 + 5i)]
= (12 + 15i + 8i + 10i^2) / (16 + 20i - 20i - 25i^2)
= (12 + 23i + 10(-1)) / (16 - 25(-1))
= (12 + 23i - 10) / (16 + 25)
= 2 + 23i / 41
= 2/41 + (23/41)i
So, the answer is (2/41) + (23/41)i, which matches your result.
In conclusion, the answers for the first and third expressions are correct, but the answer for the second expression is incorrect.