2. Simplify without the use of a calculator: $$ \begin{array}{ll} ext { a) } \sqrt{-4}+\sqrt{-16}-\sqrt{81} & ext { b) }-\sqrt{-9}-\sqrt{-25}+\sqrt{-64}-\sqrt{-1} \ ext { c) } \sqrt{-3}+\sqrt{-48} & ext { d) } \sqrt{-12}+\sqrt{-36}+\sqrt{-28} \ ext { e) } \frac{\sqrt{-18}}{\sqrt{-2}}+\sqrt{-144} & \end{array} $$

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2. Simplify without the use of a calculator: $$ \begin{array}{ll}	ext { a) } \sqrt{-4}+\sqrt{-16}-\sqrt{81} & 	ext { b) }-\sqrt{-9}-\sqrt{-25}+\sqrt{-64}-\sqrt{-1} \ 	ext { c) } \sqrt{-3}+\sqrt{-48} & 	ext { d) } \sqrt{-12}+\sqrt{-36}+\sqrt{-28} \ 	ext { e) } \frac{\sqrt{-18}}{\sqrt{-2}}+\sqrt{-144} & \end{array} $$

UpStudy AI · Accepted Answer

Let's simplify each expression by expressing the square roots of negative numbers in terms of the imaginary unit $$ i $$, where $$ i = \sqrt{-1} $$.

### a) $$ \sqrt{-4} + \sqrt{-16} - \sqrt{81} $$

1. **Simplify each term:**
   - $$ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i $$
   - $$ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i $$
   - $$ \sqrt{81} = 9 $$ (since 81 is positive)

2. **Combine the terms:**
   $$
   2i + 4i - 9 = 6i - 9
   $$
   
   **Final Answer:** $$ 6i - 9 $$ or $$ -9 + 6i $$

---

### b) $$ -\sqrt{-9} - \sqrt{-25} + \sqrt{-64} - \sqrt{-1} $$

1. **Simplify each term:**
   - $$ -\sqrt{-9} = -\sqrt{9} \cdot i = -3i $$
   - $$ -\sqrt{-25} = -\sqrt{25} \cdot i = -5i $$
   - $$ \sqrt{-64} = \sqrt{64} \cdot i = 8i $$
   - $$ -\sqrt{-1} = -i $$

2. **Combine the terms:**
   $$
   -3i - 5i + 8i - i = (-3 - 5 + 8 - 1)i = (-4)i = -4i
   $$
   
   **Final Answer:** $$ -4i $$

---

### c) $$ \sqrt{-3} + \sqrt{-48} $$

1. **Simplify each term:**
   - $$ \sqrt{-3} = \sqrt{3} \cdot i = \sqrt{3}i $$
   - $$ \sqrt{-48} = \sqrt{16 \times 3} \cdot i = 4\sqrt{3}i $$

2. **Combine the terms:**
   $$
   \sqrt{3}i + 4\sqrt{3}i = 5\sqrt{3}i
   $$
   
   **Final Answer:** $$ 5\sqrt{3}i $$

---

### d) $$ \sqrt{-12} + \sqrt{-36} + \sqrt{-28} $$

1. **Simplify each term:**
   - $$ \sqrt{-12} = \sqrt{4 \times 3} \cdot i = 2\sqrt{3}i $$
   - $$ \sqrt{-36} = \sqrt{36} \cdot i = 6i $$
   - $$ \sqrt{-28} = \sqrt{4 \times 7} \cdot i = 2\sqrt{7}i $$

2. **Combine the terms:**
   $$
   2\sqrt{3}i + 6i + 2\sqrt{7}i = (6 + 2\sqrt{3} + 2\sqrt{7})i
   $$
   
   **Final Answer:** $$ (6 + 2\sqrt{3} + 2\sqrt{7})i $$

---

### e) $$ \frac{\sqrt{-18}}{\sqrt{-2}} + \sqrt{-144} $$

1. **Simplify each term:**
   - $$ \sqrt{-18} = \sqrt{9 \times 2} \cdot i = 3\sqrt{2}i $$
   - $$ \sqrt{-2} = \sqrt{2}i $$
   - $$ \sqrt{-144} = \sqrt{144} \cdot i = 12i $$

2. **Simplify the fraction:**
   $$
   \frac{3\sqrt{2}i}{\sqrt{2}i} = 3 \quad \text{(since $$ \sqrt{2} $$ and $$ i $$ cancel out)}
   $$

3. **Combine the terms:**
   $$
   3 + 12i
   $$
   
   **Final Answer:** $$ 3 + 12i $$

---

**Summary of All Parts:**

a) $$ -9 + 6i $$  
b) $$ -4i $$  
c) $$ 5\sqrt{3}i $$  
d) $$ (6 + 2\sqrt{3} + 2\sqrt{7})i $$  
e) $$ 3 + 12i $$
a) $$ -9 + 6i $$ b) $$ -4i $$ c) $$ 5\sqrt{3}i $$ d) $$ (6 + 2\sqrt{3} + 2\sqrt{7})i $$ e) $$ 3 + 12i $$

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