Question
Find the indicated root. \( \sqrt[3]{n^{75}} \) \( n \)
Ask by Crawford Greene. in the United States
Feb 28,2025
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Tutor-Verified Answer
Answer
The cube root of \( n^{75} \) is \( n^{25} \).
Solution
To find the cube root of n⁷⁵, recall that the cube root of any expression is the same as raising that expression to the 1/3 power. In other words, we have:
∛(n⁷⁵) = (n⁷⁵)^(1/3).
When you multiply exponents, you get:
(n⁷⁵)^(1/3) = n^(75/3) = n²⁵.
Thus, the cube root of n⁷⁵ is n²⁵.
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Simplify this solution Bonus Knowledge
To find the cube root of \( n^{75} \), we can use the property of exponents that states \( \sqrt[3]{a^b} = a^{b/3} \). Applying this property, we have: \[ \sqrt[3]{n^{75}} = n^{75/3} = n^{25} \] So, the indicated root is \( n^{25} \). Now, to make it a bit more exciting, let's consider that powers and roots can take us on a wild mathematical adventure! For instance, every exponent has its own unique behavior: when multiplying like bases, you add the exponents, and when dividing like bases, you subtract them! It's like a mathematical dance where the numbers swing and swirl in perfect harmony. Isn't it fun to explore these relationships?
