Let's solve the problem step by step.

The expression to simplify is:

$\sqrt{81} + 5\sqrt{27}$

Step 1: Simplify the square roots

• $\sqrt{81} = 9$, because $9^2 = 81$.
• For $5\sqrt{27}$, notice that $27 = 9 \cdot 3$, so: $\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}.$ Thus: $5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3}.$

Step 2: Combine terms

The simplified expression becomes: $\sqrt{81} + 5\sqrt{27} = 9 + 15\sqrt{3}.$

Final Answer:

$9 + 15\sqrt{3}$

Would you like further clarification or details about this process? Let me know!

Here are 5 related questions for further practice:

1. Simplify $\sqrt{64} + 3\sqrt{32}$.
2. Simplify $2\sqrt{50} - 3\sqrt{18}$.
3. Expand and simplify $(2 + \sqrt{3})(3 - \sqrt{3})$.
4. Combine and simplify $4\sqrt{27} + 2\sqrt{12}$.
5. Rationalize the denominator of $\frac{5}{\sqrt{3}}$.

Tip: When simplifying square roots, always look for perfect square factors to simplify them systematically.