1. **Stating the problem:** Simplify the expression $$\left(\sqrt{7x + 1}\right)^3$$ given the condition $$x \geq \frac{1}{7}$$. 2. **Understanding the expression:** The expression is $$\left(\sqrt{7x + 1}\right)^3$$ which means the square root of $$7x + 1$$ raised to the third power. 3. **Recall the property of radicals and exponents:** $$\sqrt{a} = a^{\frac{1}{2}}$$, so $$\left(\sqrt{7x + 1}\right)^3 = \left((7x + 1)^{\frac{1}{2}}\right)^3$$. 4. **Use the power of a power rule:** $$\left(a^{m}\right)^n = a^{m \cdot n}$$, so $$\left((7x + 1)^{\frac{1}{2}}\right)^3 = (7x + 1)^{\frac{1}{2} \times 3} = (7x + 1)^{\frac{3}{2}}$$. 5. **Final simplified form:** $$\boxed{(7x + 1)^{\frac{3}{2}}}$$. 6. **Domain check:** Since the original expression involves a square root, the radicand must be non-negative: $$7x + 1 \geq 0 \implies x \geq -\frac{1}{7}$$. Given the problem states $$x \geq \frac{1}{7}$$, this domain is valid and satisfies the radicand condition. --- **Answer:** $$\left(\sqrt{7x + 1}\right)^3 = (7x + 1)^{\frac{3}{2}}$$ for $$x \geq \frac{1}{7}$$.