1. **State the problem:** A group of friends is renting a beach house for 9000. They want to split the cost equally, but one friend suggests that each person's share should be equal to the square root of the number of people times 1000. 2. **Define variables:** Let $n$ be the number of friends sharing the rent. 3. **Set up the equation:** Each person's share is $\sqrt{n} \times 1000$. Since the total rent is 9000, the sum of all shares is: $$n \times (\sqrt{n} \times 1000) = 9000$$ 4. **Simplify the equation:** $$n \times \sqrt{n} \times 1000 = 9000$$ $$1000 n \sqrt{n} = 9000$$ Divide both sides by 1000: $$n \sqrt{n} = 9$$ 5. **Rewrite $n \sqrt{n}$:** Since $\sqrt{n} = n^{1/2}$, then $$n \sqrt{n} = n \times n^{1/2} = n^{3/2}$$ So the equation becomes: $$n^{3/2} = 9$$ 6. **Solve for $n$:** Raise both sides to the power $\frac{2}{3}$ to isolate $n$: $$n = 9^{\frac{2}{3}}$$ 7. **Calculate $9^{2/3}$:** First, write 9 as $3^2$: $$9^{2/3} = (3^2)^{2/3} = 3^{4/3}$$ This is: $$3^{1 + 1/3} = 3^1 \times 3^{1/3} = 3 \times \sqrt[3]{3}$$ Approximate $\sqrt[3]{3} \approx 1.442$: $$n \approx 3 \times 1.442 = 4.326$$ Since the number of friends must be a whole number, we check nearby integers. 8. **Check integer values:** - For $n=4$: $$4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$$ - For $n=5$: $$5^{3/2} = (\sqrt{5})^3 \approx (2.236)^3 = 11.18$$ The value 9 lies between 8 and 11.18, so the closest integer is 4 or 5. 9. **Interpretation:** Since the problem likely expects an integer number of friends, the number of friends sharing the rent is approximately 4 or 5. The exact solution is $n = 9^{2/3}$. **Final answer:** $$\boxed{n = 9^{\frac{2}{3}} \approx 4.33}$$ This means about 4 to 5 friends are sharing the rent under the suggested payment scheme.