Answer:
1. leave the radical symbol and don't try to convert it to a decimal form
2. The approximated value differs from the exact solution and doesn't give a true equation.
3. Depending on the case, if she needs a mathematical answer, the exact value should be used, but for more practical applications, the rounded decimal form would be more usable.
Step-by-step explanation:
The step by step solution to the equation:
![4x^3=756\\x^3=\frac{756}{4} =189\\x=\sqrt[3]{189} \\x=\sqrt[3]{3^3*7} \\x=3*\sqrt[3]{7}](https://tex.z-dn.net/?f=4x%5E3%3D756%5C%5Cx%5E3%3D%5Cfrac%7B756%7D%7B4%7D%20%3D189%5C%5Cx%3D%5Csqrt%5B3%5D%7B189%7D%20%5C%5Cx%3D%5Csqrt%5B3%5D%7B3%5E3%2A7%7D%20%5C%5Cx%3D3%2A%5Csqrt%5B3%5D%7B7%7D)
1.- Exact solution means that if in the final step when solving for x the value of 
is not a perfect cube, one needs to leave it indicated as a radical expression (with the radical symbol).
in our case, the cubic root of 189 is not a perfect cube. The factor form of 189 is: 
, so there is a perfect cube factor (
), but the other factor (7) is a prime number. Therefore 3 can get out of the root, while 7 stays inside.
2.- The equation was solved above, in exact form. Now to solve it giving a decimal approximation, we use a calculator to find the cubic root of 7, which is an irrational number with infinite number of decimals, the first of which we type here: ![\sqrt[3]{7} = 1.91293118...](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B7%7D%20%3D%201.91293118...)
Therefore, the decimal approximation to the solving for x would be:
![x=3*\sqrt[3]{7}=3*1,91293118...=5.73879...=5.74](https://tex.z-dn.net/?f=x%3D3%2A%5Csqrt%5B3%5D%7B7%7D%3D3%2A1%2C91293118...%3D5.73879...%3D5.74)
Where we rounded to two decimals as requested.
When we replace the exact answer in the original expression, we get a perfect equality:
![4*x^3=756\\4* (3\sqrt[3]{7} )^3=756\\4*3^3*(\sqrt[3]{7} )^3=756\\4*27*7=756\\\\756=756](https://tex.z-dn.net/?f=4%2Ax%5E3%3D756%5C%5C4%2A%20%283%5Csqrt%5B3%5D%7B7%7D%20%29%5E3%3D756%5C%5C4%2A3%5E3%2A%28%5Csqrt%5B3%5D%7B7%7D%20%29%5E3%3D756%5C%5C4%2A27%2A7%3D756%5C%5C%5C%5C756%3D756)
While if we use the approximate answer, we get:
which is NOT a true equality.
3.- I would stick with the idea of showing the exact answer as answer to the mathematical equation. but for a practical case (for example she needs to by some material as a result of her equation solving, it would be more practical to take the numerical approximation to the store, instead of a cubic root of a number.
