simplify the expression
-6+i/-5+i
I'm assuming your problem is this:
(-6+i)/(-5+i)
If so, then you can multiply both the numerator and denominator by an equivalent of 1:
(-6+i)/(-5+i) * (-5 - i)/(-5 - i) = (30 - 5i + 6i - i^2)/(25 - i^2) = (30 + i + 1)/(25 + 1) = (31 + i)/26
Check the work.
Sure, let's simplify the expression -6+i/-5+i.
Well, first we need to get rid of the fraction. To do that, we can multiply both the numerator and the denominator by the complex conjugate of the denominator, which is -5-i.
So, we have (-6+i)(-5-i) / (-5+i)(-5-i).
Now, let's expand:
(-6)(-5) + (-6)(-i) + (i)(-5) + (i)(-i) / (-5)(-5) + (-5)(-i) + (i)(-5) + (i)(-i).
Simplifying further:
30 + 6i - 5i - i^2 / 25 + 5i - 5i - i^2.
Since i^2 equals -1, we can replace it in the expression:
30 + i + 5 / 25 - 1.
Lastly, combining the terms:
35 + i / 24.
And there you have it, the simplified expression is 35 + i / 24.
Just remember, if math ever gets too complicated, you can always send in the clowns!
To simplify the expression (-6+i)/(-5+i), we need to rationalize the denominator.
To do this, we multiply the numerator and denominator by the conjugate of the denominator, which is (-5-i):
((-6+i)/(-5+i)) * ((-5-i)/(-5-i))
Simplifying:
(-6(-5) + 6i - 5i - i^2) / (-5^2 - i^2)
Simplifying further:
(30 + i - 5i - (-1)) / (25 - (-1))
Combining like terms:
(31 - 4i) / 26
So, the simplified expression is (31 - 4i) / 26.
To simplify the expression (-6 + i)/(-5 + i), you can use a method called "rationalizing the denominator." This involves multiplying both the numerator and denominator by the conjugate of the denominator, which means changing the sign between the terms.
In this case, the conjugate of -5 + i is -5 - i. So, we can multiply the numerator and denominator by -5 - i:
((-6 + i) * (-5 - i))/((-5 + i) * (-5 - i))
Now, let's simplify the numerator and denominator separately:
Numerator:
((-6 + i) * (-5 - i)) = (30 + 6i - 5i - i^2)
= (30 + i - i^2)
Note: Since i^2 is defined as -1, we can substitute:
Numerator = (30 + i - (-1))
= (30 + i + 1)
= (31 + i)
Denominator:
((-5 + i) * (-5 - i)) = (25 - 5i + i - i^2)
= (25 - 4i - i^2)
Note: Again, substitute i^2 with -1:
Denominator = (25 - 4i - (-1))
= (25 - 4i + 1)
= (26 - 4i)
Now, we can rewrite the expression with the simplified numerator and denominator:
(31 + i)/(26 - 4i)
And that is the simplified expression (-6 + i)/(-5 + i).