Simplify 54−−√3 .
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To simplify 54^(1/3):
Since the cube root of 54 is not an integer, we can write it as a simplified radical expression.
Let's find the prime factorization of 54:
54 = 2 × 3 × 3 × 3 = 2 × 3^3
Now we can write 54^(1/3) as (2 × 3^3)^(1/3):
Using the property (a × b)^(1/n) = a^(1/n) × b^(1/n), we can simplify it as follows:
(2 × 3^3)^(1/3) = 2^(1/3) × (3^3)^(1/3)
Since (3^3)^(1/3) = 3, we have:
2^(1/3) × (3^3)^(1/3) = 2^(1/3) × 3 = 3 × 2^(1/3)
Hence, 54^(1/3) simplifies to 3 × 2^(1/3).