Question
Rational and Irrational Numbers Quick Check 2 of 52 of 5 Items Question Which of the following is irrational?(1 point) Responses 3√4 Start Fraction Start Root 3 End Root over 4 End Fraction 3.6363636363636363... 3.6363636363636363... 52.781654292 52.781654292 −7837 negative 7 Start Fraction 8 over 37 End Fraction
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GPT-4o mini
To determine which of the given options is irrational, let's analyze each one:
1. **3√4** - This means the cube root of 4. The cube root of any rational number is also rational, so this is rational.
2. **Start Fraction Start Root 3 End Root over 4 End Fraction** - This represents \(\frac{\sqrt{3}}{4}\). Since \(\sqrt{3}\) is an irrational number, \(\frac{\sqrt{3}}{4}\) is also irrational.
3. **3.6363636363636363...** - This is a repeating decimal, which makes it a rational number.
4. **52.781654292** - This is a non-repeating, non-terminating decimal. However, if it has a finite number of decimal places, it is rational. Since we don't know if it continues indefinitely or not from the representation, but typically, this would be treated as rational.
5. **−7837** - This is a whole number and therefore rational.
6. **Start Fraction 8 over 37 End Fraction** - This fraction represents \(\frac{8}{37}\), which is a rational number.
Given this breakdown, the option that is irrational is:
**Start Fraction Start Root 3 End Root over 4 End Fraction** (or \(\frac{\sqrt{3}}{4}\)).
1. **3√4** - This means the cube root of 4. The cube root of any rational number is also rational, so this is rational.
2. **Start Fraction Start Root 3 End Root over 4 End Fraction** - This represents \(\frac{\sqrt{3}}{4}\). Since \(\sqrt{3}\) is an irrational number, \(\frac{\sqrt{3}}{4}\) is also irrational.
3. **3.6363636363636363...** - This is a repeating decimal, which makes it a rational number.
4. **52.781654292** - This is a non-repeating, non-terminating decimal. However, if it has a finite number of decimal places, it is rational. Since we don't know if it continues indefinitely or not from the representation, but typically, this would be treated as rational.
5. **−7837** - This is a whole number and therefore rational.
6. **Start Fraction 8 over 37 End Fraction** - This fraction represents \(\frac{8}{37}\), which is a rational number.
Given this breakdown, the option that is irrational is:
**Start Fraction Start Root 3 End Root over 4 End Fraction** (or \(\frac{\sqrt{3}}{4}\)).
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