simplify the expression
-6+i/-5+i
Could someone explain the steps?
recall that (a+bi)(a-bi) = a^2+b^2
to get that pesky complex number out of the denominator, multiply by (-5-i) top and bottom:
(-6+i)(-5-i)/(-5-i)(-5+i)
= (30 - 5i + 6i - i^2)/26
= (31 + i)/26
= 31/26 + 1/26 i
thank you
Sure! Let me attempt to simplify the expression for you, but with a touch of humor!
Now, we have the expression -6 + i over -5 + i.
To simplify this, we want to eliminate the complex "i" in the denominator. And what better way to eliminate problems than with a little bit of humor?
So, let's multiply both the numerator and the denominator by the conjugate of the denominator, which is -5 - i.
But before we proceed, let me warn you – dealing with complex numbers can be a little tricky. It's like juggling bowling pins while riding a unicycle, but I'm here to guide you through the circus!
Alright, let's multiply -6 + i with -5 - i, and -5 + i with -5 - i. Remember, when you multiply a complex number by its conjugate, you end up with the sum of squares – it's like a mathematical magic trick!
(-6 + i) times (-5 - i) = (-6 * -5) + (-6 * -i) + (i * -5) + (i * -i), you follow?
Simplifying that, we get 30 + 6i - 5i + i^2. Pssst, don't forget that i^2 is equal to -1, like a clown forgetting his own name!
Now, combining like terms, we have 30 + i + i^2 - 5i. And since we know that i^2 is -1, we can simplify it further to 30 + i - 5i - 1.
Now, group the real numbers and imaginary numbers together – it's like bringing all the clowns into one tiny car!
Combining the real numbers, we have 30 - 1, which is 29, and combining the imaginary numbers gives us -5i + i, which is -4i.
So, after all the mathematic acrobatics, the simplified expression is 29 - 4i.
And there you have it! 29 - 4i, the answer to your expression, delivered with a dash of humor. Thank you for attending the Clown Bot's Math Circus!
To simplify the expression (-6+i)/(-5+i), follow these steps:
Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of (-5+i) is (-5-i). So, multiply the numerator and denominator by (-5-i):
(-6+i) * (-5-i) / (-5+i) * (-5-i)
Step 2: Simplify using the FOIL method in the numerator and the difference of squares in the denominator.
The numerator (-6+i) * (-5-i) gives: 30 + 11i - i^2
Simplifying, we get: 30 + 11i + 1 (since i^2 = -1)
Therefore, the numerator is: 31 + 11i
The denominator (-5+i) * (-5-i) gives: 25 - i^2
Simplifying, we get: 25 + 1 (since i^2 = -1)
Therefore, the denominator is: 26
Step 3: Write the simplified expression.
The simplified expression is: (31 + 11i) / 26
To simplify the expression (-6 + i) / (-5 + i), we can use a technique called complex conjugates.
Step 1: Take the conjugate of the denominator.
The conjugate of -5 + i is -5 - i.
Step 2: Multiply the numerator and denominator by the conjugate of the denominator.
(-6 + i) * (-5 - i) / (-5 + i) * (-5 - i)
Step 3: Simplify the expression.
Multiplying the two complex numbers in the numerator gives
30 - 6i -5i + i^2
Multiplying the two complex numbers in the denominator gives
25 - i^2
Note that i^2 is equal to -1, so we have:
30 - 6i - 5i - 1 / 25 - (-1)
Simplifying further, we get:
29 - 11i / 26
So, the simplified expression is 29 - 11i / 26.