Write the following expression as a complex number in standard form.
2-7i/-3+8i
multiply top and bottom by (-3 - 8i) to get
(2-7i)(-3-8i) / (-3+8i)(-3-8i)
(-6 + 21i - 16i + 56i^2) / (9 + 64)
(-62 + 5i)/73
To write the expression (2-7i)/(-3+8i) as a complex number in standard form, we need to rationalize the denominator. Rationalizing the denominator means getting rid of the imaginary number in the denominator by multiplying both the numerator and denominator by its conjugate (the same expression with the imaginary part negated).
The conjugate of -3+8i is -3-8i. So, we multiply the numerator and denominator by -3-8i:
[(2-7i)(-3-8i)] / [(-3+8i)(-3-8i)]
Now, let's simplify the numerator:
(2 * -3) + (2 * -8i) + (-7i * -3) + (-7i * -8i)
-6 - 16i + 21i + 56i²
Since i² is defined as -1, we can replace i² with -1:
-6 - 16i + 21i + 56(-1)
-6 - 16i + 21i - 56
-62 + 5i
Now, let's simplify the denominator:
(-3 * -3) + (-3 * -8i) + (8i * -3) + (8i * -8i)
9 + 24i - 24i - 64i²
Again, replacing i² with -1:
9 + 24i - 24i - 64(-1)
9 + 24i - 24i + 64
73
Finally, we have:
(-6 - 16i + 21i + 56i²) / (9 + 24i - 24i - 64i²)
which simplifies to:
(-62 + 5i) / 73
Therefore, the expression (2-7i)/(-3+8i) in standard form is:
-62/73 + (5/73)i