Question

Simplify the expression fraction with numerator of the square root of negative one and denominator of the quantity three plus eight times i minus the quantity two plus five times i.

fraction with numerator x-3 plus i and denominator 10
fraction with numerator negative 3 plus i and denominator 10
fraction with numerator 3 minus i and denominator 10
fraction with numerator negative 3 minus i and denominator 10

Answer 1

The expression is simplified by using the imaginary unit i and removing the imaginary number from the denominator, which yields \( \frac{3+i}{10} \).

Step-by-step explanation:

To simplify the expression we start with the square root of negative one, which is known as the imaginary unit i. Let's begin by simplifying the denominator:
(3 + 8i) - (2 + 5i)
= (3 - 2) + (8i - 5i)
= 1 + 3i
Now we can write the original expression as:
\( \frac{i}{1 + 3i} \)

To remove the imaginary number from the denominator, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is 1 - 3i:
\( \frac{i}{1 + 3i} \times \frac{1 - 3i}{1 - 3i} \)
= \( \frac{i(1 - 3i)}{1^2 - (3i)^2} \)
= \( \frac{i - 3i^2}{1 - 9(-1)} \)
Since i^2 equals -1, this further simplifies to:
= \( \frac{i + 3}{1 + 9} \)
= \( \frac{i + 3}{10} \)

Therefore, the simplified expression is \( \frac{3+i}{10} \), which corresponds to option three from the listed answers.