1. We are given the equation $$\sqrt[4]{8} \cdot 625 \cdot 81 \cdot \Box = \Box \cdot 5 \cdot 2$$ where \Box represents unknown values that are the same on both sides. 2. To solve for \Box, first simplify known terms: - $$\sqrt[4]{8} = 8^{1/4} = (2^3)^{1/4} = 2^{3/4}$$ - $$625 = 5^4$$ - $$81 = 3^4$$ - On the right side, $$5\cdot 2 = 10$$ 3. Write the equation substituting simplified terms: $$2^{3/4} \cdot 5^4 \cdot 3^4 \cdot \Box = \Box \cdot 10$$ 4. Divide both sides by \Box (since it is nonzero): $$2^{3/4} \cdot 5^4 \cdot 3^4 = 10$$ 5. Evaluate the left side to check for consistency: - $$5^4 = 625$$ - $$3^4 = 81$$ So the left side is: $$2^{3/4} \cdot 625 \cdot 81$$ 6. Calculate numeric value: - $$2^{3/4} = \sqrt[4]{8} \approx 1.68179$$ - Multiply: $$1.68179 \cdot 625 = 1051.118$$ - Multiply by 81: $$1051.118 \cdot 81 = 85140.56$$ Left side is approximately 85140.56 but right side is 10. 7. The only way for the equation to hold is if \Box on the left side is a number that when multiplied by approx 85140.56 gives the same as \Box on right side multiplied by 10. Let the unknowns be \Box_{left} and \Box_{right} but problem states same \Box on both sides. So: $$85140.56 \cdot \Box = 10 \cdot \Box$$ 8. The only solution for this to be true for nonzero \Box is impossible unless \Box = 0. 9. Alternatively, if \Box represents two different unknowns, solve for \Box on one side: Let \Box = x on left and \Box = y on right: $$2^{3/4} \cdot 625 \cdot 81 \cdot x = y \cdot 10$$ Assuming \Box is the same value, the problem asks for \Box that satisfies: $$2^{3/4} \cdot 625 \cdot 81 \cdot \Box = \Box \cdot 10$$ Divide both sides by \Box: $$2^{3/4} \cdot 625 \cdot 81 = 10$$ Which is false, so \Box must be zero. 10. Therefore, the only solution is $$\Box = 0$$. **Final answer:** $$\boxed{0}$$