To solve the problem, let us work through each part of the expression:

Expression to evaluate:

$8 \sin(60^\circ) + 18 \tan(30^\circ)$

Step 1: Recall the trigonometric values

1. $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
2. $\tan(30^\circ) = \frac{1}{\sqrt{3}}$

Step 2: Substitute the values into the expression

$8 \sin(60^\circ) + 18 \tan(30^\circ) = 8 \left(\frac{\sqrt{3}}{2}\right) + 18 \left(\frac{1}{\sqrt{3}}\right)$

Simplify each term: $8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}, \quad 18 \times \frac{1}{\sqrt{3}} = \frac{18}{\sqrt{3}}.$

Step 3: Simplify the second term further

Rationalize $\frac{18}{\sqrt{3}}$: $\frac{18}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = 6\sqrt{3}.$

Step 4: Combine the terms

$4\sqrt{3} + 6\sqrt{3} = 10\sqrt{3}.$

Final Answer:

$\boxed{10\sqrt{3}}$

Let me know if you'd like further clarification or have additional questions!

5 Related Questions:

1. What are the exact values of other standard trigonometric ratios like $\cos(30^\circ)$ and $\tan(60^\circ)$?
2. How do you rationalize a denominator in fractions involving square roots?
3. Can you derive the values of $\sin(60^\circ)$ and $\tan(30^\circ)$ using a 30-60-90 triangle?
4. How would this problem change if $\tan(30^\circ)$ was replaced by $\cos(30^\circ)$?
5. How can trigonometric identities help simplify similar expressions?

Tip:

Always memorize the key trigonometric values of angles like $30^\circ$, $45^\circ$, and $60^\circ$, as they are frequently used in problems.