Final answer:
To express square roots as mixed radicals, find the largest perfect square factor of the number and simplify. √172 is expressed as 2√43, while √86, √43, and √21 do not simplify since they do not have perfect square factors other than 1.
Step-by-step explanation:
The question asks to express square roots as mixed radicals. A mixed radical is a way of expressing a square root that contains a whole number (if possible) and a square root of a non-square number. We need to find the largest perfect square factor of the given numbers and express it in mixed radical form.
- For √172: We can express 172 as 4 × 43, where 4 is a perfect square. Therefore, √172 = √(4 × 43) = 2√43.
- For √86: 86 doesn't have a perfect square factor other than 1, so it remains as is, √86.
- For √43: Similar to 86, 43 also does not have a perfect square factor other than 1, so it remains as is, √43.
- For √21: Again, 21 does not have a perfect square factor other than 1, so it remains as is, √21.