Here are the solutions for simplifying the radicals in the problems:

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

To simplify a cube root:

• $32 = 2^5$, so $\sqrt[3]{32} = 2 \cdot \sqrt[3]{2^2}$.
• $g^9$ simplifies as $\sqrt[3]{g^9} = g^3$.
• $h^8$ simplifies as $\sqrt[3]{h^8} = h^2 \cdot \sqrt[3]{h^2}$.

Solution:
[ \sqrt[3]{32g^9h^8} = 2g^3h^2 \sqrt[3]{4h^2} ]

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Problem 13: $(16x^{12}y^8)^{\frac{1}{4}}$

Apply the fourth root to each term:

• $16 = 2^4$, so $\sqrt[4]{16} = 2$.
• $x^{12}$: Divide the exponent by 4, $x^{12/4} = x^3$.
• $y^8$: Divide the exponent by 4, $y^{8/4} = y^2$.

Solution:
$(16x^{12}y^8)^{\frac{1}{4}} = 2x^3y^2$

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Problem 14: $5m^5 \cdot m^{\frac{2}{5}} \cdot \left(3m^4\right)^4$

Simplify step by step:

• $5m^5 \cdot m^{\frac{2}{5}} = 5m^{5 + \frac{2}{5}} = 5m^{\frac{27}{5}}$.
• $(3m^4)^4 = 3^4 \cdot m^{16} = 81m^{16}$.
  Combine terms:
• $5m^{\frac{27}{5}} \cdot 81m^{16} = 5 \cdot 81 \cdot m^{\frac{27}{5} + 16}$.
  Convert $16$ to fifths: $16 = \frac{80}{5}$, so:
  $\frac{27}{5} + \frac{80}{5} = \frac{107}{5}.$

Solution:
$5m^5 \cdot m^{\frac{2}{5}} \cdot (3m^4)^4 = 405m^{\frac{107}{5}}$

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

Simplify the fraction under the square root:

• $\frac{16}{4} = 4$.
• For $x^5 / x^3$, subtract exponents: $x^{5-3} = x^2$.
  $\sqrt{\frac{16x^5}{4x^3}} = \sqrt{4x^2} = 2x$

Solution:
$\sqrt{\frac{16x^5}{4x^3}} = 2x$

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

Simplify under the square root:

• For $a$: $a^{4-8} = a^{-4}$.
• For $b$: $b^{7-3} = b^4$.
• For $c$: $c^{9-3} = c^6$.

Take the square root:

• $a^{-4}$ becomes $a^{-2}$.
• $\sqrt{b^4} = b^2$.
• $\sqrt{c^6} = c^3$.

Solution:
$\sqrt{\frac{a^4b^7c^9}{a^8b^3c^3}} = \frac{b^2c^3}{a^2}$

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

Simplify the fraction under the square root:

• $\frac{36}{9} = 4$.
• For $m$: $m^{2-6} = m^{-4}$.
• For $n$: $n^{7-1} = n^6$.

Take the square root:

• $\sqrt{4} = 2$.
• $m^{-4}$ becomes $m^{-2}$.
• $\sqrt{n^6} = n^3$.

Solution:
$\sqrt{\frac{36m^2n^7}{9m^6n}} = \frac{2n^3}{m^2}$

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Summary of Solutions:

1. $\sqrt[3]{32g^9h^8} = 2g^3h^2 \sqrt[3]{4h^2}$
2. $(16x^{12}y^8)^{\frac{1}{4}} = 2x^3y^2$
3. $5m^5 \cdot m^{\frac{2}{5}} \cdot (3m^4)^4 = 405m^{\frac{107}{5}}$
4. $\sqrt{\frac{16x^5}{4x^3}} = 2x$
5. $\sqrt{\frac{a^4b^7c^9}{a^8b^3c^3}} = \frac{b^2c^3}{a^2}$
6. $\sqrt{\frac{36m^2n^7}{9m^6n}} = \frac{2n^3}{m^2}$

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Do you want a deeper explanation for any problem? Let me know!

Related Questions:

1. How do you simplify cube roots of exponents in general?
2. How do you work with fractional exponents?
3. How do you simplify radical expressions with variables?
4. How do you combine terms with different bases and exponents?
5. What happens when you encounter negative exponents under a radical?

Tip: When simplifying radicals with fractions, always simplify the numerator and denominator separately before applying the radical.