Problem 37 - Algebra and Trigonometry

Question

Perform the indicated operations and write the result in standard form. $$ \frac{-8+\sqrt{-32}}{24} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{-8+\sqrt{-32}}{24} \) is \( \frac{-1+\frac{1}{2}\sqrt{2}i}{3} \).

Step 1: Understand the Imaginary Unit

In mathematics, when there is a square root of a negative number, the result is a complex value which comprises a real part and an imaginary part. The square root of -1 is represented by the imaginary unit \( i \). Hence, the square root of any negative number can be defined in terms of \( i \). The square root of -32 can be written as \( \sqrt{-32} = \sqrt{32}i = 4\sqrt{2}i \).

Step 2: Simplify the Numerator

Substitute \( 4\sqrt{2}i \) back into the numerator of the fraction, the numerator will be as follows: \( -8+4\sqrt{2}i \).

Step 3: Simplify the Fraction

The complete fraction after substitution is \( \frac{-8+4\sqrt{2}i}{24} \). Now divide every term in the numerator and the denominator by 8. Hence, the fraction simplifies to \( \frac{-1+\frac{1}{2}\sqrt{2}i}{3} \), which is the standard form of the complex number.