1. **Stating the problem:** Given the equation $$\sqrt[7]{a} = 27 (a x^{\frac{15}{7}})$$, we want to understand or simplify this expression. 2. **Recall the definition of the 7th root:** $$\sqrt[7]{a} = a^{\frac{1}{7}}$$. 3. **Rewrite the equation using exponents:** $$a^{\frac{1}{7}} = 27 \cdot a \cdot x^{\frac{15}{7}}$$ 4. **Express all terms with exponents:** $$a^{\frac{1}{7}} = 27 \cdot a^{1} \cdot x^{\frac{15}{7}}$$ 5. **Divide both sides by $$a^{\frac{1}{7}}$$ to isolate terms:** $$1 = 27 \cdot a^{1 - \frac{1}{7}} \cdot x^{\frac{15}{7}} = 27 \cdot a^{\frac{6}{7}} \cdot x^{\frac{15}{7}}$$ 6. **Rewrite the equation:** $$27 \cdot a^{\frac{6}{7}} \cdot x^{\frac{15}{7}} = 1$$ 7. **If solving for $$x$$, divide both sides by $$27 a^{\frac{6}{7}}$$:** $$x^{\frac{15}{7}} = \frac{1}{27 a^{\frac{6}{7}}}$$ 8. **Raise both sides to the power $$\frac{7}{15}$$ to solve for $$x$$:** $$x = \left( \frac{1}{27 a^{\frac{6}{7}}} \right)^{\frac{7}{15}} = \frac{1^{\frac{7}{15}}}{27^{\frac{7}{15}} \cdot a^{\frac{6}{7} \cdot \frac{7}{15}}} = \frac{1}{27^{\frac{7}{15}} \cdot a^{\frac{6}{15}}}$$ 9. **Simplify the exponent on $$a$$:** $$\frac{6}{15} = \frac{2}{5}$$ 10. **Final expression for $$x$$:** $$x = \frac{1}{27^{\frac{7}{15}} \cdot a^{\frac{2}{5}}}$$ **Summary:** We expressed the original equation in terms of exponents and solved for $$x$$ in terms of $$a$$.