Evaluate $\sqrt[4]{81}$
Answer (2)
The problem asks to evaluate 4 81 .
Recognize that 81 = 3 4 .
Therefore, 4 81 = 4 3 4 .
Thus, the answer is 3 .
Explanation
Understanding the problem We are asked to evaluate the fourth root of 81, which means we need to find a number that, when raised to the fourth power, equals 81.
Expressing the problem mathematically We can express this mathematically as finding x such that x 4 = 81 .
Finding the solution We know that 3 4 = 3 × 3 × 3 × 3 = 9 × 9 = 81 . Therefore, the fourth root of 81 is 3.
Final Answer Thus, 4 81 = 3 .
Examples
Understanding roots and exponents is crucial in many fields, such as engineering and physics. For example, when calculating the resonant frequency of an LC circuit, you use the formula f = 2 π L C 1 , where you need to find the square root of the product of inductance (L) and capacitance (C). Similarly, in mechanical engineering, when analyzing vibrations, you might encounter equations involving square roots or higher-order roots to determine natural frequencies or damping coefficients. These calculations help engineers design stable and efficient systems.
The fourth root of 81 is 3 because 81 can be expressed as 3 4 . Therefore, 4 81 = 4 3 4 = 3 .
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