Questions: Perform the indicated operations. [ frac-7-sqrt-1842 ]

Solution Steps

To solve the given expression, we need to simplify the square root of a negative number using imaginary numbers. Then, perform the arithmetic operations to simplify the fraction.

The expression involves the square root of a negative number, \(\sqrt{-18}\). This can be expressed using imaginary numbers as \(\sqrt{-18} = \sqrt{18} \cdot i\), where \(i\) is the imaginary unit. Calculating \(\sqrt{18}\), we have: \[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \approx 4.2426 \] Thus, \(\sqrt{-18} \approx 4.2426i\).

Substitute \(\sqrt{-18} \approx 4.2426i\) into the original expression: \[ \frac{-7 - \sqrt{-18}}{42} = \frac{-7 - 4.2426i}{42} \]

Divide both the real and imaginary parts of the numerator by the denominator: \[ \frac{-7}{42} = -0.1667 \] \[ \frac{-4.2426i}{42} \approx -0.1010i \] Thus, the expression simplifies to: \[ -0.1667 - 0.1010i \]

Final Answer