Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( -18-\sqrt{-69} \)
Ask by Perkins Christensen. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(-18 - i \sqrt{69}\)
Solution
To express the given expression \( -18 - \sqrt{-69} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Identify the Imaginary Part:**
The term \( \sqrt{-69} \) involves the square root of a negative number, which can be expressed using the imaginary unit \( i \):
\[
\sqrt{-69} = i \sqrt{69}
\]
2. **Rewrite the Expression:**
Substitute the imaginary part back into the original expression:
\[
-18 - \sqrt{-69} = -18 - i \sqrt{69}
\]
3. **Simplify if Possible:**
The radical \( \sqrt{69} \) cannot be simplified further since 69 factors into \( 3 \times 23 \), neither of which are perfect squares.
Thus, the expression in its simplified complex form is:
\[
-18 - i \sqrt{69}
\]
**Answer:**
\(\boxed{-18\ -\ i\,\sqrt{69}}\)
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Simplify this solution Mind Expander
First, we can rewrite \( -\sqrt{-69} \) using the imaginary unit \( i \). This gives us \( -\sqrt{-69} = -\sqrt{69} \cdot i \). Therefore, we can rewrite the entire expression as: \[ -18 - \sqrt{-69} = -18 - \sqrt{69} \cdot i \] Now, we have a complex number where the real part is \( -18 \) and the imaginary part is \( -\sqrt{69} \). Thus, the expression simplifies to: \[ -18 - \sqrt{69} \cdot i \] In the standard form of a complex number \( a + bi \), this can be written as: \[ -18 + (-\sqrt{69})i \]
