To simplify the expression \( \frac{4}{1 - 3^{1/3} + 9^{1/3}} \), we will start by rewriting \( 9^{1/3} \):
\[ 9^{1/3} = (3^2)^{1/3} = 3^{2/3} \]
This allows us to rewrite the expression in the denominator:
\[ 1 - 3^{1/3} + 9^{1/3} = 1 - 3^{1/3} + 3^{2/3} \]
Now let's let \( x = 3^{1/3} \). This means that \( 3^{2/3} = x^2 \). Substituting \( x \) into the denominator, we have:
\[ 1 - x + x^2 \]
So, we can rewrite the original expression as:
\[ \frac{4}{1 - x + x^2} \]
Next, we will simplify \( 1 - x + x^2 \). We can factor this quadratic expression. To factor \( 1 - x + x^2 \), we can observe that the discriminant may help us identify if it can be factored nicely:
The discriminant \( D \) of the quadratic \( ax^2 + bx + c \) is given by
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = -1 \), and \( c = 1 \):
\[ D = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]
Since the discriminant is negative, \( 1 - x + x^2 \) cannot be factored into real linear factors. We can complete the square:
\[ 1 - x + x^2 = \left(x - \frac{1}{2}\right)^2 + \frac{3}{4} \]
Now the denominator can be expressed as:
\[ 1 - x + x^2 = \left( x - \frac{1}{2} \right)^2 + \frac{3}{4} \]
Next, we need to put this back into the fraction:
\[ \frac{4}{\left( x - \frac{1}{2} \right)^2 + \frac{3}{4}} \]
Since we have:
\[ x = 3^{1/3} \]
We can note that the original expression doesn't seem to simplify much further. A better approach would be just to leave it as:
\[ \frac{4}{1 - 3^{1/3} + 3^{2/3}} \]
However, we can confirm that the simplified rational form given the transformations we have made will be indeed:
\[ \frac{4}{1 - 3^{1/3} + 3^{2/3}} \]
If you want an approximate numerical evaluation, \( 3^{1/3} \approx 1.442 \), and hence:
\[ 1 - 3^{1/3} + 3^{2/3} \approx 1 - 1.442 + 2.08 \approx 1.638 \]
Thus, the approximate value would yield:
\[ \frac{4}{1.638} \approx 2.44 \]
But for algebraic simplification, we stick to:
\[ \frac{4}{1 - 3^{1/3} + 3^{2/3}} \]
is a simplified expression.