Here are the solutions for the math problems shown in the image:

12. $\sqrt[3]{32g^9h^8}$

1. Factorize each term:
   • $32 = 2^5$
   • $g^9$ and $h^8$ already have exponents.
2. Apply the cube root:
   • $\sqrt[3]{2^5} = 2^{5/3} = 2^1 \cdot \sqrt[3]{2^2}$
   • $\sqrt[3]{g^9} = g^{9/3} = g^3$
   • $\sqrt[3]{h^8} = h^{8/3} = h^2 \cdot \sqrt[3]{h^2}$
3. Combine terms: [ \sqrt[3]{32g^9h^8} = 2g^3h^2 \cdot \sqrt[3]{4h^2} ]

13. $(16x^{12}y^4)^{1/4}$

1. Distribute the power $1/4$ to each term:
   • $16^{1/4} = 2$ (since $16 = 2^4$)
   • $(x^{12})^{1/4} = x^{12/4} = x^3$
   • $(y^4)^{1/4} = y^{4/4} = y$
2. Combine: $(16x^{12}y^4)^{1/4} = 2x^3y$

14. $5m^2 \cdot m^5 \cdot (3n^4)^{3/4}$

1. Combine like terms for $m$:
   • $m^2 \cdot m^5 = m^{2+5} = m^7$
2. Simplify $(3n^4)^{3/4}$:
   • $3^{3/4}$ stays as is.
   • $(n^4)^{3/4} = n^{4 \cdot 3/4} = n^3$
3. Combine everything: $5m^2 \cdot m^5 \cdot (3n^4)^{3/4} = 5m^7 \cdot 3^{3/4}n^3$ Final result: $5m^7 \cdot 3^{3/4}n^3$

15. $\sqrt{\frac{16x^5}{4x^3}}$

1. Simplify the fraction inside the square root:
   • $\frac{16}{4} = 4$
   • $\frac{x^5}{x^3} = x^{5-3} = x^2$
2. Apply the square root:
   • $\sqrt{4} = 2$
   • $\sqrt{x^2} = x$
3. Combine: $\sqrt{\frac{16x^5}{4x^3}} = 2x$

16. $\sqrt{\frac{a^4b^7c^9}{a^8b^3c^3}}$

1. Simplify the exponents:
   • $\frac{a^4}{a^8} = a^{4-8} = a^{-4}$
   • $\frac{b^7}{b^3} = b^{7-3} = b^4$
   • $\frac{c^9}{c^3} = c^{9-3} = c^6$
2. Apply the square root:
   • $\sqrt{a^{-4}} = a^{-2} = \frac{1}{a^2}$
   • $\sqrt{b^4} = b^2$
   • $\sqrt{c^6} = c^3$
3. Combine: $\sqrt{\frac{a^4b^7c^9}{a^8b^3c^3}} = \frac{b^2c^3}{a^2}$

17. $\sqrt{\frac{36m^2n^7}{9m^6n}}$

1. Simplify the fraction:
   • $\frac{36}{9} = 4$
   • $\frac{m^2}{m^6} = m^{2-6} = m^{-4}$
   • $\frac{n^7}{n} = n^{7-1} = n^6$
2. Apply the square root:
   • $\sqrt{4} = 2$
   • $\sqrt{m^{-4}} = m^{-2} = \frac{1}{m^2}$
   • $\sqrt{n^6} = n^3$
3. Combine: $\sqrt{\frac{36m^2n^7}{9m^6n}} = \frac{2n^3}{m^2}$

Final Answers:

1. 12: $2g^3h^2 \cdot \sqrt[3]{4h^2}$
2. 13: $2x^3y$
3. 14: $5m^7 \cdot 3^{3/4}n^3$
4. 15: $2x$
5. 16: $\frac{b^2c^3}{a^2}$
6. 17: $\frac{2n^3}{m^2}$

Would you like detailed explanations for any specific part?

Here are 5 related questions you might want to explore:

1. How do you simplify radical expressions with exponents?
2. What are the rules for dividing terms with the same base?
3. How does a fractional exponent relate to roots (e.g., $x^{1/3}$)?
4. How do you simplify cube roots of variables?
5. What happens when you have negative exponents?

Tip: Always simplify the fraction or expression inside the root first before applying the radical!