Evaluate $125^{1 / 3}$
Answer (2)
Rewrite 125 as 5 3 .
Substitute 5 3 into the expression: ( 5 3 ) 1/3 .
Apply the power of a power rule: ( a m ) n = a mn .
Simplify to find the answer: 5 .
Explanation
Understanding the problem We are asked to evaluate 12 5 1/3 . This expression is asking: what number, when raised to the power of 3, equals 125? In other words, we are looking for the cube root of 125.
Rewriting 125 We can rewrite 125 as a power of a prime number. Since 5 × 5 = 25 and 25 × 5 = 125 , we can write 125 = 5 3 .
Substitution Now we can substitute 5 3 for 125 in the original expression: 12 5 1/3 = ( 5 3 ) 1/3
Applying the power of a power rule Using the power of a power rule, which states that ( a m ) n = a m × n , we can simplify the expression: ( 5 3 ) 1/3 = 5 3 × ( 1/3 ) = 5 1 = 5
Final Answer Therefore, 12 5 1/3 = 5 .
Examples
Understanding cube roots is essential in various fields, such as engineering and physics, where volumes and scaling are involved. For instance, if you're designing a cubic container that needs to hold 125 cubic meters of liquid, finding the cube root of 125 tells you the length of each side of the container. In this case, each side would be 5 meters long, ensuring the container meets the volume requirement. This concept extends to more complex scenarios, like scaling models or calculating growth rates, making it a fundamental skill in practical applications.
To evaluate 12 5 1/3 , we rewrite 125 as 5 3 and then apply the power of a power rule. The result is 5 , meaning the cube root of 125 is 5. Therefore, 12 5 1/3 = 5 .
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