1. Problem: Complete the blanks in the expressions involving powers and roots linked to the number 5 and 625. 2. Given: - $(5^3) \square$ - $5^{\square 4}$ - $625$ - $625^{\square 1}$ - $(\square 3 \sqrt{125})^{\square 4}$ 3. First, recognize that $625 = 5^4$ since $5^4 = 625$. 4. Solve for the left expression $(5^3) \square$: Since $5^3 = 125$ and the blank should multiply it to become $625$, then the blank must be 5, because $125 \times 5 = 625$. 5. Solve for $5^{\square 4}$: To equal $625 = 5^4$, the value here must be $1$ because $5^{1 \times 4} = 5^4 = 625$. 6. $625^{\square 1}$: Since exponent is on 625, to keep it $625$, exponent must be 1. 7. $(\square 3 \sqrt{125})^{\square 4}$: - Simplify $\sqrt{125} = \sqrt{5^3} = 5^{3/2}$. - If the blank before 3 is $1$, then inside root is $\sqrt{125}=5^{3/2}$. - So $(1 \cdot 3 \sqrt{125})^{\square 4} = (\sqrt{125})^{\square 4}$. - If the exponent blank is $2$, then $(5^{3/2})^{2} = 5^{3}$. But to match 625 which is $5^4$, so the expression inside should be $5$, and raised to 4 equals $5^4 = 625$. Let the blank before 3 be $\frac{1}{3}$ to turn $\sqrt[3]{125} = 5$. Then $(1 \sqrt[3]{125})^{4} = 5^{4} = 625$. Thus, blanks: first before 3 is $1$, root is cube root (3), and exponent is 4. Summary of blanks: - $(5^3) \square = (5^3) 5$ - $5^{\square 4} = 5^{1 \times 4}$ - $625^{\square 1} = 625^1$ - $(\square 3 \sqrt{125})^{\square 4} = (\sqrt[3]{125})^{4}$ Final values filled: - Left blank: 5 - Top exponent multiplier blank: 1 - Bottom exponent blank: 1 - Left inside root blank: empty (cube root symbol 3) - Right exponent blank: 4 Answer: 5, 1, 1, (cube root 3), 4