Comparing Radicals: Cube Root of 11, Square Root of 7, and Fourth Root of 45

Determining which of the expressions sqrt[3]{11}, 2sqrt{7}, and 4sqrt[4]{45} is the largest involves a detailed calculation of each value. This article comprehensively analyzes and compares these radicals, providing a clear understanding of their relative magnitudes.

Understanding the Radicals

The three expressions in question are:

The cube root of 11, denoted as sqrt[3]{11} The square root of 7, denoted as 2sqrt{7} The fourth root of 45, denoted as 4sqrt[4]{45}

To accurately compare these expressions, we need to calculate each one using a calculator or mathematical approximation methods.

Calculations

Cube Root of 11

To find the value of the cube root of 11, we use a calculator:

sqrt[3]{11} approx 2.2247

Multiplying 2.2247 by 3 gives:

3 times 2.2247 approx 6.6741

Square Root of 7

The square root of 7, when doubled, is calculated as follows:

sqrt{7} approx 2.6458

Multiplying 2.6458 by 2 gives:

2 times 2.6458 approx 5.2916

Fourth Root of 45

The fourth root of 45 can be calculated by finding the fourth root of 45 directly or using the approximation:

45^{1/4}  (3^2 cdot 5^{1/4})  (3^{1/2} cdot 5^{1/4}) approx 1.7321 cdot 1.4953 approx 2.5858

Multiplying 2.5858 by 4 gives:

4 times 2.5858 approx 10.3432

Conclusion

Summarizing the approximate values, we have:

The cube root of 11 is approximately 6.6741 The square root of 7, doubled, is approximately 5.2916 The fourth root of 45, multiplied by 4, is approximately 10.3432

Based on these calculations, it is clear that the largest value is 4th root of 45, which is approximately 10.3432.

Additional Insights

It's interesting to note that the second expression, 2nd root of 7, when doubled, is approximately 2.64575131, making it the largest among the three when considering the original expression as a single entity. However, when we double the square root of 7, we obtain approximately 5.2916, which is smaller than the values obtained for the cube root of 11 and the fourth root of 45 when multiplied by their respective coefficients.

For an even more detailed analysis, we can substitute the values back into the original expressions to confirm:

3rd root of 11 2.22 2nd root of 7 2.65 4th root of 45 2.59

Thus, the square root of 7 is indeed the largest value among the original expressions, but when doubled, it is still smaller than the cube root of 11 and the fourth root of 45 when multiplied by their respective coefficients.