Let's solve each problem step-by-step.

Problem 13

$\sqrt{50a^6bc^9}$

1. Simplify the square root of each term:

   • For $50$, break it down as $50 = 25 \cdot 2$, so $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
   • For $a^6$, since the exponent is even, $\sqrt{a^6} = a^{6/2} = a^3$.
   • For $b$, it remains under the square root as it does not have an even exponent.
   • For $c^9$, split it into $c^8 \cdot c$, so $\sqrt{c^9} = \sqrt{c^8 \cdot c} = c^4\sqrt{c}$.

2. Combine the results: $\sqrt{50a^6bc^9} = 5a^3c^4\sqrt{2bc}$

Problem 15

$\sqrt[3]{27a^3b^2c^{12}}$

1. Simplify the cube root of each term:

   • For $27$, since $27 = 3^3$, we have $\sqrt[3]{27} = 3$.
   • For $a^3$, the cube root is $a^{3/3} = a$.
   • For $b^2$, leave it under the cube root as it does not divide evenly by 3.
   • For $c^{12}$, the cube root is $c^{12/3} = c^4$.

2. Combine the results: [ \sqrt[3]{27a^3b^2c^{12}} = 3ac^4\sqrt[3]{b^2} ]

Final Answers

1. $\sqrt{50a^6bc^9} = 5a^3c^4\sqrt{2bc}$
2. $\sqrt[3]{27a^3b^2c^{12}} = 3ac^4\sqrt[3]{b^2}$

Would you like a deeper explanation on any of these steps?

Related Questions

1. How do you simplify higher roots, such as the fourth or fifth root?
2. What are the rules for simplifying radicals with variables?
3. How can you handle complex numbers when simplifying radicals?
4. How do fractional exponents relate to radicals?
5. How do you rationalize the denominator of a radical expression?

Tip:

When simplifying radicals with variables, focus on splitting exponents into multiples of the root degree to make simplification easier.