Compare the following irrational numbers
2pi? 3 square root 66
Answers
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2π > 3√66
To compare the irrational numbers 2π and √(66), we can use numerical or algebraic methods.
Numerically:
1. Calculate the decimal approximation of 2π using a calculator: 2π ≈ 6.283185307179586.
2. Calculate the decimal approximation of √(66) using a calculator: √(66) ≈ 8.12403840463596.
3. Compare the two decimal approximations: 6.283185307179586 < 8.12403840463596.
Algebraically:
1. Recognize that 2π is approximately 6.28, which is approximately equal to 6.
2. Identify that the square root of 66 is greater than 8, since √(64) = 8 and 66 is larger than 64.
3. Conclude that 2π < √(66).
Therefore, both methods lead to the same conclusion: 2π < √(66).
To compare the irrational numbers 2π and 3√66, we can follow these steps:
1. Convert both irrational numbers into decimal approximations.
- The value of π is approximately 3.14159.
- To find the decimal approximation of √66, use a calculator or a math software program. The approximate value of √66 is about 8.12404.
2. Compare the decimal approximations.
- Comparing the decimal approximations, we have:
2π ≈ 2 * 3.14159 = 6.28318
3√66 ≈ 3 * 8.12404 = 24.37212
3. Compare the decimal values.
- From the decimal approximations, we see that 6.28318 is less than 24.37212.
Therefore, we can conclude that 2π is less than 3√66, which can be represented as:
2π < 3√66.