Question: User: $\frac{\sqrt{10}+\sqrt{10}+\sqrt{10}}{\sqrt{30}}$ = $81^{x^{3}}$ x=0.5
1. **State the problem:** Simplify the expression $$\frac{\sqrt{10}+\sqrt{10}+\sqrt{10}}{\sqrt{30}}$$ and verify the equality with $$81^{x^{3}}$$ where $$x=0.5$$.
2. **Simplify the numerator:** Since there are three identical terms $$\sqrt{10}$$ added together, we have:
$$\sqrt{10}+\sqrt{10}+\sqrt{10} = 3\sqrt{10}$$
3. **Rewrite the expression:**
$$\frac{3\sqrt{10}}{\sqrt{30}}$$
4. **Simplify the fraction:**
$$\frac{3\sqrt{10}}{\sqrt{30}} = 3 \times \frac{\sqrt{10}}{\sqrt{30}} = 3 \times \sqrt{\frac{10}{30}}$$
5. **Simplify inside the square root:**
$$\sqrt{\frac{10}{30}} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}$$
6. **Substitute back:**
$$3 \times \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}}$$
7. **Rationalize the denominator:**
$$\frac{3}{\sqrt{3}} = \frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{3} = \cancel{\frac{\cancel{3}\sqrt{3}}{\cancel{3}}} = \sqrt{3}$$
8. **Evaluate the right side:**
Given $$x=0.5$$, calculate $$81^{x^{3}}$$:
First, compute $$x^{3} = (0.5)^{3} = 0.125$$
Then,
$$81^{0.125} = (3^{4})^{0.125} = 3^{4 \times 0.125} = 3^{0.5} = \sqrt{3}$$
9. **Conclusion:** Both sides equal $$\sqrt{3}$$, so the equality holds true.
**Final answer:** $$\frac{\sqrt{10}+\sqrt{10}+\sqrt{10}}{\sqrt{30}} = 81^{(0.5)^{3}} = \sqrt{3}$$