Simplify $(x^{\frac{1}{2}})^{\frac{1}{6}}$.
Answer (2)
The expression ( x 2 1 ) 6 1 simplifies to x 12 1 by applying the power of a power rule and multiplying the exponents. The final result is x 12 1 .
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Apply the power of a power rule: ( a m ) n = a m × n .
Multiply the exponents: 2 1 × 6 1 = 12 1 .
Simplify the expression: x 12 1 .
The simplified expression is x 12 1 .
Explanation
Understanding the Problem We are given the expression ( x 2 1 ) 6 1 . Our goal is to simplify it.
Applying the Power of a Power Rule To simplify the expression, we need to use the power of a power rule, which states that when you raise a power to another power, you multiply the exponents: ( a m ) n = a m × n
Multiplying the Exponents In our case, we have ( x 2 1 ) 6 1 . Applying the power of a power rule, we get: x 2 1 × 6 1
Calculating the Resulting Exponent Now, we multiply the exponents: 2 1 × 6 1 = 12 1
Final Answer Therefore, the simplified expression is: x 12 1
Examples
Understanding exponent rules is crucial in many scientific fields. For instance, in physics, when dealing with wave functions or quantum mechanics, you often encounter expressions with exponents. Simplifying these expressions correctly helps in solving complex equations and understanding the behavior of particles. Similarly, in computer science, exponents are used in algorithms and data structures to optimize performance and manage memory efficiently. Knowing how to manipulate exponents allows for better resource allocation and faster computation times.
