1. **Problem:** Find all possible roots of the radicals given. 2. **Recall:** The nth root of a real number $a$ is written as $$\sqrt[n]{a}$$. If no index is shown, it is assumed to be 2 (square root). 3. **Important rules:** - Even roots of positive numbers have two roots: a positive and a negative root. - Odd roots of positive numbers have one positive root. - Odd roots of negative numbers have one negative root. - Even roots of negative numbers are not real. - The radical sign indicates the principal (positive) root. 4. **Calculate each:** - $\sqrt{16} = \pm 4$ because $4^2=16$ and $(-4)^2=16$ - $\sqrt{121} = \pm 11$ - $\sqrt{289} = \pm 17$ - $\sqrt[4]{\frac{4}{25}} = \pm \frac{\sqrt{2}}{5}$ since $\left(\frac{\sqrt{2}}{5}\right)^4 = \frac{4}{25}$ - $\sqrt[3]{8} = 2$ (only one root, cube root of positive 8) - $\sqrt[3]{343} = 7$ - $\sqrt[3]{-125} = -5$ - $\sqrt[3]{\frac{1}{27}} = \frac{1}{3}$ - $\sqrt{1} = \pm 1$ - $\sqrt[4]{2401} = \pm 7$ because $7^4=2401$ - $\sqrt[4]{4096} = \pm 8$ - $\sqrt[4]{\frac{81}{16}} = \pm \frac{3}{2}$ 5. **Table of roots:** - Even index, positive radicand: 2 roots (positive and negative) - Odd index, positive radicand: 1 positive root - Odd index, negative radicand: 1 negative root - Even index, negative radicand: No real roots 6. **Solve problems 1 to 4:** 1. $\sqrt{117}$: Since 117 is positive and index 2 (even), roots are $\pm \sqrt{117}$. 2. $4\sqrt{320}$: Simplify inside root first: $$\sqrt{320} = \sqrt{64 \times 5} = 8\sqrt{5}$$ So, $$4 \times 8 \sqrt{5} = 32 \sqrt{5}$$ 3. $2\sqrt[3]{48}$: Cube root of 48: $$\sqrt[3]{48} = \sqrt[3]{16 \times 3} = \sqrt[3]{16} \times \sqrt[3]{3}$$ Since $16 = 2^4$, cube root of $2^4 = 2^{4/3} = 2^{1 + 1/3} = 2 \times 2^{1/3}$ So, $$\sqrt[3]{48} = 2 \times 2^{1/3} \times \sqrt[3]{3} = 2 \times \sqrt[3]{6}$$ Therefore, $$2 \times 2 \times \sqrt[3]{6} = 4 \sqrt[3]{6}$$ 4. $3\sqrt[3]{108}$: Simplify inside root: $$108 = 27 \times 4$$ $$\sqrt[3]{108} = \sqrt[3]{27} \times \sqrt[3]{4} = 3 \times \sqrt[3]{4}$$ Multiply by 3: $$3 \times 3 \times \sqrt[3]{4} = 9 \sqrt[3]{4}$$ 7. **Additional problems:** 5. $\sqrt[3]{-250}$: $$-250 = -125 \times 2$$ $$\sqrt[3]{-250} = \sqrt[3]{-125} \times \sqrt[3]{2} = -5 \sqrt[3]{2}$$ 6. $6 \sqrt[3]{-2}$: Cube root of -2 is $-\sqrt[3]{2}$, so $$6 \times (-\sqrt[3]{2}) = -6 \sqrt[3]{2}$$ 7. $3 \sqrt[4]{62}$: 62 is positive, so $$3 \sqrt[4]{62}$$ (cannot simplify further) 8. $5 \sqrt[4]{2592}$: Factor 2592: $$2592 = 16 \times 162 = 16 \times 81 \times 2$$ $$\sqrt[4]{2592} = \sqrt[4]{16} \times \sqrt[4]{81} \times \sqrt[4]{2} = 2 \times 3 \times \sqrt[4]{2} = 6 \sqrt[4]{2}$$ Multiply by 5: $$5 \times 6 \sqrt[4]{2} = 30 \sqrt[4]{2}$$ 8. **Radicals with variables:** - Square roots: exponents must be multiples of 2 to simplify fully. - Cube roots: exponents must be multiples of 3. - Fourth roots: exponents must be multiples of 4. 9. $\sqrt{32x^1y^9}$: $$32 = 16 \times 2$$ $$\sqrt{32x^1y^9} = \sqrt{16} \times \sqrt{2} \times \sqrt{x} \times \sqrt{y^9} = 4 \sqrt{2} \times \sqrt{x} \times y^{4} \sqrt{y} = 4 y^{4} \sqrt{2xy}$$ 10. $\sqrt{324a^3b^7}$: $$324 = 18^2$$ $$\sqrt{324a^3b^7} = 18 \times a^{1} \sqrt{a} \times b^{3} \sqrt{b} = 18 a b^{3} \sqrt{a b}$$ 11. $\sqrt[3]{216m^3n^6}$: $$216 = 6^3$$ $$\sqrt[3]{216m^3n^6} = 6 m n^{2}$$ 12. $\sqrt[3]{56r^8s^4}$: $$56 = 8 \times 7$$ $$\sqrt[3]{56r^8s^4} = \sqrt[3]{8} \times \sqrt[3]{7} \times r^{2} \sqrt[3]{r^{2}} \times s \sqrt[3]{s} = 2 r^{2} s \sqrt[3]{7 r^{2} s}$$ 13. $\sqrt[3]{64x^{10}y^{21}}$: $$64 = 4^3$$ $$\sqrt[3]{64x^{10}y^{21}} = 4 x^{3} x \times y^{7} = 4 x^{3} y^{7} \sqrt[3]{x}$$ 14. $\sqrt[3]{-81p^{2}q^{12}}$: $$-81 = -27 \times 3$$ $$\sqrt[3]{-81p^{2}q^{12}} = \sqrt[3]{-27} \times \sqrt[3]{3} \times p^{0} \sqrt[3]{p^{2}} \times q^{4} = -3 q^{4} \sqrt[3]{3 p^{2}}$$ 15. $\sqrt{5v^{5}}$: $$\sqrt{5} \times v^{2} \sqrt{v} = v^{2} \sqrt{5 v}$$ 16. $\sqrt[4]{48m^{8}n^{3}}$: $$48 = 16 \times 3$$ $$\sqrt[4]{48m^{8}n^{3}} = \sqrt[4]{16} \times \sqrt[4]{3} \times m^{2} \sqrt[4]{n^{3}} = 2 m^{2} n^{3/4} \sqrt[4]{3}$$ 17. $\sqrt[4]{625e^{23}d^{11}}$: $$625 = 5^{4}$$ $$\sqrt[4]{625e^{23}d^{11}} = 5 e^{5} e^{3/4} d^{2} d^{3/4} = 5 e^{5} d^{2} \sqrt[4]{e^{3} d^{3}}$$ 18. $\sqrt[4]{(v+3)^{8}} = (v+3)^{2}$ **Final answers are shown above with all intermediate steps.**