Sure, let's break down the problem step by step.
First, we will address arranging the given numbers in ascending order of magnitude:
- The numbers to be arranged are [tex]\(\sqrt[3]{3}, \sqrt[3]{4}, \sqrt[3]{2}\)[/tex].
To solve this, we evaluate these cube roots:
1. [tex]\(\sqrt[3]{2}\)[/tex] approximately equals [tex]\(1.2599210498948732\)[/tex],
2. [tex]\(\sqrt[3]{3}\)[/tex] approximately equals [tex]\(1.4422495703074083\)[/tex],
3. [tex]\(\sqrt[3]{4}\)[/tex] approximately equals [tex]\(1.5874010519681994\)[/tex].
Now, we sort these values from smallest to largest:
- [tex]\(1.2599210498948732\)[/tex] corresponding to [tex]\(\sqrt[3]{2}\)[/tex],
- [tex]\(1.4422495703074083\)[/tex] corresponding to [tex]\(\sqrt[3]{3}\)[/tex],
- [tex]\(1.5874010519681994\)[/tex] corresponding to [tex]\(\sqrt[3]{4}\)[/tex].
Therefore, the ascending order is:
1. [tex]\(\sqrt[3]{2}\)[/tex],
2. [tex]\(\sqrt[3]{3}\)[/tex],
3. [tex]\(\sqrt[3]{4}\)[/tex].
Next, we will address arranging the given numbers in descending order of magnitude:
- The numbers to be arranged are [tex]\(\sqrt[8]{90}, \sqrt[7]{10}, \sqrt{6}\)[/tex].
To solve this, we evaluate these roots:
1. [tex]\(\sqrt{6}\)[/tex] (square root of 6) approximately equals [tex]\(2.449489742783178\)[/tex],
2. [tex]\(\sqrt[7]{10}\)[/tex] (seventh root of 10) approximately equals [tex]\(1.3894954943731377\)[/tex],
3. [tex]\(\sqrt[8]{90}\)[/tex] (eighth root of 90) approximately equals [tex]\(1.7550129025853407\)[/tex].
Now, we sort these values from largest to smallest:
- [tex]\(2.449489742783178\)[/tex] corresponding to [tex]\(\sqrt{6}\)[/tex],
- [tex]\(1.7550129025853407\)[/tex] corresponding to [tex]\(\sqrt[8]{90}\)[/tex],
- [tex]\(1.3894954943731377\)[/tex] corresponding to [tex]\(\sqrt[7]{10}\)[/tex].
Therefore, the descending order is:
1. [tex]\(\sqrt{6}\)[/tex],
2. [tex]\(\sqrt[8]{90}\)[/tex],
3. [tex]\(\sqrt[7]{10}\)[/tex].
So, summarizing:
- The ascending order for [tex]\(\sqrt[3]{3}, \sqrt[3]{4}, \sqrt[3]{2}\)[/tex] is: [tex]\(\sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{4}\)[/tex].
- The descending order for [tex]\(\sqrt[8]{90}, \sqrt[7]{10}, \sqrt{6}\)[/tex] is: [tex]\(\sqrt{6}, \sqrt[8]{90}, \sqrt[7]{10}\)[/tex].
