To solve the expression [tex]\(\frac{7}{5 + \sqrt[3]{2}}\)[/tex], we'll break down the steps in detail:
1. Identify the numerator and the denominator: In the given expression, the numerator is 7, and the denominator is [tex]\(5 + \sqrt[3]{2}\)[/tex].
2. Simplification: Our target is to handle the denominator [tex]\(5 + \sqrt[3]{2}\)[/tex]. Understanding the cube root of 2 can be tricky as it is an irrational number.
3. Combine the components: Although we could explore multiple algebraic techniques to work with the cube root, the goal is to handle it in such a way that cleans up the expression as much as possible. But given the complexity due to the irrational number and simple addition in the denominator, it's better to perform a detailed evaluation numerically rather than purely algebraically.
4. Evaluate the Expressions Numerically:
- Compute [tex]\(5 + \sqrt[3]{2}\)[/tex] precisely.
- [tex]\( \sqrt[3]{2}\)[/tex] is approximately 1.259921.
- So, [tex]\(5 + \sqrt[3]{2} \approx 5 + 1.259921 \approx 6.259921\)[/tex].
- Next, calculate the fraction [tex]\(\frac{7}{6.259921}\)[/tex].
5. Final Calculation:
- By dividing [tex]\(7\)[/tex] by [tex]\(6.259921\)[/tex], we find:
[tex]\[
\frac{7}{6.259921} \approx 1.118225.
\][/tex]
Therefore, the precise estimate of [tex]\(\frac{7}{5 + \sqrt[3]{2}}\)[/tex] is approximately [tex]\(1.1182249654917862\)[/tex].
By following these steps methodically, we conclude that the value of the expression [tex]\(\frac{7}{5 + \sqrt[3]{2}}\)[/tex] is approximately [tex]\(1.1182249654917862\)[/tex].
