$$ \frac { a V } { 1 - V } = \sqrt[ 3 ] { \frac { 2 V + T } { a + 2 b } } $$

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UpStudy AI · Accepted Answer

1. Start with the equation  
   $$
   \frac{aV}{1-V} = \sqrt[3]{\frac{2V+T}{a+2b}}.
   $$

2. Cube both sides to eliminate the cube root. This gives  
   $$
   \left(\frac{aV}{1-V}\right)^3 = \frac{2V+T}{a+2b}.
   $$

3. Multiply both sides by $$a+2b$$ to clear the denominator on the right-hand side:  
   $$
   (a+2b)\left(\frac{aV}{1-V}\right)^3 = 2V+T.
   $$

4. Write out the cube on the left-hand side:  
   $$
   (a+2b) \cdot \frac{a^3 V^3}{(1-V)^3} = 2V+T.
   $$

5. Multiply both sides by $$(1-V)^3$$ to eliminate the denominator:  
   $$
   (a+2b)a^3V^3 = (2V+T)(1-V)^3.
   $$

6. Rearranging the equation leads to the polynomial form  
   $$
   (2V+T)(1-V)^3 - (a+2b)a^3 V^3 = 0.
   $$

This is a cubic equation in $$V$$. In general, to find an explicit solution for $$V$$ one would need to solve this cubic equation using standard methods for cubic equations.
To solve the equation $$ \frac{aV}{1 - V} = \sqrt[3]{\frac{2V + T}{a + 2b}} $$, follow these steps: 1. **Cube Both Sides**: $$ \left(\frac{aV}{1 - V}\right)^3 = \frac{2V + T}{a + 2b} $$ 2. **Multiply Both Sides by $$a + 2b$$**: $$ (a + 2b) \cdot \left(\frac{aV}{1 - V}\right)^3 = 2V + T $$ 3. **Expand the Left Side**: $$ (a + 2b) \cdot \frac{a^3 V^3}{(1 - V)^3} = 2V + T $$ 4. **Multiply Both Sides by $$(1 - V)^3$$**: $$ (a + 2b) a^3 V^3 = (2V + T)(1 - V)^3 $$ 5. **Rearrange the Equation**: $$ (2V + T)(1 - V)^3 - (a + 2b) a^3 V^3 = 0 $$ This results in a cubic equation in terms of $$V$$. Solving this cubic equation will give the value(s) of $$V$$.

\( \frac { a V } { 1 - V } = \sqrt[ 3 ] { \frac { 2 V + T } { a + 2 b } } \)

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