The problem asks you to simplify the given radicals by combining them using the indicated operations, assuming all variables are positive.

Given Expression:

[ \sqrt[3]{8x^{14}y^{23}} + \sqrt[3]{8x^{14}y^{23}} - \sqrt[3]{8x^{13}y^5} ]

Step-by-step solution:

1. Simplify each term under the cube root:

   • First term: $\sqrt[3]{8x^{14}y^{23}}$

     • $8 = 2^3$, so $8$ can be factored out of the cube root.
     • $x^{14} = x^{12} \cdot x^2$ (factor the largest power divisible by 3, $x^{12}$).
     • $y^{23} = y^{21} \cdot y^2$ (factor $y^{21}$, divisible by 3).
     • Simplify: [ \sqrt[3]{8x^{14}y^{23}} = 2x^4y^7 \cdot \sqrt[3]{x^2y^2} ]

   • Second term: Identical to the first term: [ \sqrt[3]{8x^{14}y^{23}} = 2x^4y^7 \cdot \sqrt[3]{x^2y^2} ]

   • Third term: $\sqrt[3]{8x^{13}y^5}$

     • $x^{13} = x^{12} \cdot x^1$ (factor $x^{12}$).
     • $y^5 = y^3 \cdot y^2$ (factor $y^3$).
     • Simplify: [ \sqrt[3]{8x^{13}y^5} = 2x^4y \cdot \sqrt[3]{xy^2} ]

2. Combine the terms:

   • Expression becomes: [ 2x^4y^7 \cdot \sqrt[3]{x^2y^2} + 2x^4y^7 \cdot \sqrt[3]{x^2y^2} - 2x^4y \cdot \sqrt[3]{xy^2} ]

   • Combine the like terms: [ (2x^4y^7 + 2x^4y^7) \cdot \sqrt[3]{x^2y^2} - 2x^4y \cdot \sqrt[3]{xy^2} ]

   • Simplify: [ 4x^4y^7 \cdot \sqrt[3]{x^2y^2} - 2x^4y \cdot \sqrt[3]{xy^2} ]

3. Final Answer: [ 4x^4y^7 \cdot \sqrt[3]{x^2y^2} - 2x^4y \cdot \sqrt[3]{xy^2} ]

Let me know if you want a further breakdown or have any questions!

Related Questions:

1. What are the rules for simplifying cube roots with exponents?
2. How do you factor terms under a cube root?
3. Can radicals with different bases ever be combined?
4. Why do we assume all variables are positive in this problem?
5. How would the solution change if we had a higher-degree root, like $\sqrt[5]{}$?

Tip:

Always break down exponents into multiples of the root's degree to simplify radicals effectively.