To simplify the given expression:

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

we will:

1. Break down each term inside the cube roots into factors for simplification.
2. Combine like terms wherever possible.

Step 1: Simplify each cube root

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

The constant $8$ can be expressed as $2^3$, so:

[ \sqrt[3]{8x^{14}y^{23}} = 2 \cdot \sqrt[3]{x^{14}y^{23}}. ]

Second term: $\sqrt[3]{x^{14}y^{23}}$

This term is already in its simplest form.

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

Here, the constant $8 = 2^3$, so:

[ \sqrt[3]{8x^{13}y^5} = 2 \cdot \sqrt[3]{x^{13}y^5}. ]

Step 2: Combine terms

Substitute the simplified forms back into the expression:

[ 2 \cdot \sqrt[3]{x^{14}y^{23}} + \sqrt[3]{x^{14}y^{23}} - 2 \cdot \sqrt[3]{x^{13}y^5}. ]

Combine like terms for $\sqrt[3]{x^{14}y^{23}}$:

[ 2 \cdot \sqrt[3]{x^{14}y^{23}} + \sqrt[3]{x^{14}y^{23}} = 3 \cdot \sqrt[3]{x^{14}y^{23}}. ]

Thus, the expression becomes:

[ 3 \cdot \sqrt[3]{x^{14}y^{23}} - 2 \cdot \sqrt[3]{x^{13}y^5}. ]

Final Simplified Expression:

[ 3 \cdot \sqrt[3]{x^{14}y^{23}} - 2 \cdot \sqrt[3]{x^{13}y^5}. ]

Would you like me to explain further, or explore how to simplify the cube roots further? Let me know!

Related Questions:

1. How do cube roots simplify for variables with exponents?
2. What happens when coefficients in front of radicals are combined?
3. How can you simplify mixed radical expressions involving cube roots?
4. Can this expression be factored further?
5. How does the assumption of positive variables affect simplification?

Tip:

Always check if terms under radicals share common factors to simplify further or combine like terms.