Which of the following is an irrational number between 5 and 7: 6–√, 12−−√, 27−−√, 36−−√?
To determine which number is irrational, we need to recall that irrational numbers cannot be expressed as fractions and do not terminate or repeat.
First, we can simplify each of the given expressions to determine if they can be expressed as fractions:
- 6–√ cannot be simplified further and does not have a rational square root.
- 12−−√ simplifies to 2√3, which can be expressed as a fraction (2√3/1).
- 27−−√ simplifies to 3√3, which can be expressed as a fraction (3√3/1).
- 36−−√ simplifies to 6, which is a rational number.
Thus, the only number that is irrational between 5 and 7 is 6–√.
To determine which of the given numbers is irrational, we need to identify which numbers cannot be expressed as a fraction or terminating decimal.
Let's simplify each option:
1. 6 - √:
This can be simplified to 6 - √1 = 5, which is not between 5 and 7.
2. 12 - √√:
This expression is unclear, as it seems to contain an extra square root (√) symbol.
3. 27 - √√:
Similar to the previous option, this expression is not clear due to the extra square root (√) symbol.
4. 36 - √:
This can be simplified to 36 - 6 = 30, which is not between 5 and 7.
Based on the simplifications, we can conclude that none of the given numbers is an irrational number between 5 and 7.