Step-by-step examples showing how to simplify radical expressions to add and subtract unlike square roots.

The image presents two examples of adding and subtracting unlike square roots, with step-by-step simplifications. Let's go through each example carefully, explain the process, and ensure clarity.

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🔷 Example 1:

$$
2\sqrt{8} + 5\sqrt{18} + 7
$$

#### Step 1: Simplify each square root.
We simplify $\sqrt{8}$ and $\sqrt{18}$ by factoring out perfect squares.

- $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$
- $\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$

Now substitute back:
$$
2(2\sqrt{2}) + 5(3\sqrt{2}) + 7
$$

#### Step 2: Multiply coefficients.
$$
(2)(2)\sqrt{2} + (5)(3)\sqrt{2} + 7 = 4\sqrt{2} + 15\sqrt{2} + 7
$$

#### Step 3: Combine like terms.
Both $4\sqrt{2}$ and $15\sqrt{2}$ have the same radical part ($\sqrt{2}$), so we add their coefficients:
$$
(4 + 15)\sqrt{2} + 7 = 19\sqrt{2} + 7
$$

✔ Final Answer:
$$
\boxed{19\sqrt{2} + 7}
$$

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🔷 Example 2:

$$
\sqrt{20} - \sqrt{45}
$$

#### Step 1: Simplify each square root.

- $\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}$
- $\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}$

Now substitute:
$$
2\sqrt{5} - 3\sqrt{5}
$$

#### Step 2: Combine like terms.
Both terms have $\sqrt{5}$, so combine coefficients:
$$
(2 - 3)\sqrt{5} = (-1)\sqrt{5} = -\sqrt{5}
$$

✔ Final Answer:
$$
\boxed{-\sqrt{5}}
$$

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✔ Summary of Key Concepts:

- Unlike square roots (e.g., $\sqrt{2}$ and $\sqrt{3}$) cannot be combined directly unless simplified.
- Always simplify radicals first by factoring out perfect squares.
- Only like radicals (same radicand under the square root) can be added or subtracted.
- After simplification, combine coefficients of like radicals.

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📌 General Rule:

$$
a\sqrt{b} + c\sqrt{b} = (a + c)\sqrt{b}
$$
But:
$$
a\sqrt{b} + c\sqrt{d} \quad \text{(if } b \ne d\text{)} \quad \text{cannot be combined unless simplified.}
$$

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Let me know if you'd like to try more practice problems!