Question

Arrange the following surds in increasing order.

1. \(\sqrt [3]{2}\)
2. \(\sqrt 3\)
3. \(\sqrt [4]{5}\)
4. \(\sqrt [6]{7}\)

• A. A < B < C < D
• B. A < B < C < D
• C. D < B < C < A
• D. A < D < C < B

Answer 1

The Correct Option is D

To arrange the given surds \(\sqrt[3]{2}\), \(\sqrt{3}\), \(\sqrt[4]{5}\), and \(\sqrt[6]{7}\) in increasing order, we need to compare their numerical values. Since these are roots with different indices, we can simplify the comparison by converting them into a common form (e.g., exponents with the same denominator) or calculating their approximate decimal values. Here, we will calculate the approximate decimal values:

1. \(\sqrt[3]{2}\): The cube root of 2 is approximately 1.2599.
2. \(\sqrt{3}\): The square root of 3 is approximately 1.7321.
3. \(\sqrt[4]{5}\): The fourth root of 5 is approximately 1.4953.
4. \(\sqrt[6]{7}\): The sixth root of 7 is approximately 1.3115.

By comparing these decimal values, we can arrange the surds as follows:

• \(\sqrt[3]{2} \approx 1.2599\)
• \(\sqrt[6]{7} \approx 1.3115\)
• \(\sqrt[4]{5} \approx 1.4953\)
• \(\sqrt{3} \approx 1.7321\)

Thus, the order of increasing surds is: \(\sqrt[3]{2} < \sqrt[6]{7} < \sqrt[4]{5} < \sqrt{3}\).

Hence, the correct order is represented by the option A < D < C < B.

Answer 2

We calculate the approximate numerical values of the surds:

\[ A = \sqrt[3]{2} \approx 1.26, \quad B = \sqrt{3} \approx 1.73, \quad C = \sqrt[4]{5} \approx 1.50, \quad D = \sqrt[6]{7} \approx 1.40 \]

Now, arranging the surds in increasing order:

\[ A \approx 1.26, \quad D \approx 1.40, \quad C \approx 1.50, \quad B \approx 1.73 \]

Thus, the increasing order is:

\[A < D < C < B\]