Evaluate fifth root of 462

Problem

√(5,462)

Solution

1. Identify the expression as the fifth root of the number 462 which can be written as 462^(1/5)

2. Check for perfect fifth powers by listing small integers raised to the fifth power: 1^5=1 2^5=32 3^5=243 and 4^5=1024

3. Observe that 243<462<1024 which means the value lies between 3 and 4

4. Factor the number 462 to see if it contains any perfect fifth power factors: 462=2×3×7×11

5. Conclude that since there are no repeated prime factors, the expression cannot be simplified further in radical form.

6. Approximate the value using a calculator or numerical methods (like Newton's method) to find the decimal representation.

Final Answer

√(5,462)≈3.4047

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