Final answer:
The expression 1/⁴√1800 simplifies to 1/30, which already has a rational denominator because the fourth root of 1800 simplifies to 30.
Step-by-step explanation:
The student has asked to express the fractional expression 1/⁴√1800 with a rational denominator. To convert this to a rational denominator, we need to rationalize the denominator. The fourth root of 1800 can be simplified before rationalizing since 1800 can be factored into perfect squares and cubes.
First, we factor 1800 to find 1800 = 2² × 3² × 5². Hence, ⁴√1800 = ⁴√(2² × 3² × 5²) = 2 × 3 × 5 = 30. So, our original expression 1/⁴√1800 equals 1/30. No further steps are needed because 1/30 already has a rational denominator.
To express the fraction 1/⁴√1800 with a rational denominator, we need to simplify the radical. First, find the prime factorization of 1800: 1800 = 2 * 2 * 2 * 3 * 3 * 5 * 5. Next, rewrite the original expression as 1/⁴√(2 * 2 * 2 * 3 * 3 * 5 * 5).
Since the fourth root of each factor can be written as a fraction with a rational exponent, we can simplify further. Simplifying the expression, we get 1/(2 * 3 * ⁵√2 * 3 * 5). Rearranging the terms, we have 1/(2 * 3 * ⁵√2 * ⁵√3 * ⁵√5).