The number 54 can be divided by the perfect cube 27 (27 * 2 = 54). Evaluating the cube root of 27 gives 3, so the expression simplifies to 3*cbrt(2).

Look for a perfect cube factor of 16, which is 8. Writing cbrt(16) as cbrt(8 * 2) allows us to bring out the cube root of 8 (which is 2), yielding 2*cbrt(2).

Factor 20 into a perfect square and another number, which is 4 * 5. Taking the square root of 4 gives 2, which leaves us with the simplified form 2*sqrt(5).

The number 48 can be factored into 16 * 3. Because 16 is a perfect square, its square root (4) can be brought outside the radical, resulting in 4*sqrt(3).

Find the highest perfect square factor of 72, which is 36. Since 72 = 36 * 2, we take the square root of 36 to get 6 outside, leaving 6*sqrt(2).

The largest perfect square that divides 27 is 9. We can write sqrt(27) as sqrt(9 * 3). The square root of 9 is 3, making the simplified form 3*sqrt(3).

To simplify sqrt(50), find the largest perfect square factor of 50, which is 25. Rewrite sqrt(50) as sqrt(25 * 2), which simplifies to 5*sqrt(2).

The cube root of 1000 requires finding the base that creates 1000 when multiplied by itself three times. 10 * 10 * 10 equals 1000, so the answer is 10.

To evaluate the cube root of 512, we identify the number that gives 512 when cubed. Since 8 * 8 * 8 = 512, the answer is 8.

The cube root of 343 is a number that, when raised to the third power, yields 343. That number is 7, because 7^3 = 343.