Final answer:
To simplify (16/81)³/⁴ × √100/81, raise (16/81) to the power of 3/4 to get (2/3), and find the square root of 100/81 to get (10/9). Multiply these results to find the simplified expression, which is 20/27.
Step-by-step explanation:
The student has asked to simplify the expression (16/81)³/⁴ × √100/81. To solve this, we need to apply exponent rules and simplify square roots. First, simplifying (16/81)³/⁴ involves taking the fourth root of the cube of 16/81. This can be done by raising the fraction to the power of 3 and then taking the fourth root, which is the same as raising it to the power of 3/4. To simplify √100/81, we find the square root of both the numerator and the denominator.
Let's break it down step by step:
(16/81)³ equals (2/3)³ because 16 is 2 to the fourth power and 81 is 3 to the fourth power. Thus (2/3)³ equals 8/27.
We then take the fourth root of 8/27, which is the same as raising it to the power of 1/4. (8/27)¹/⁴ equals 2/3, because 8 is 2 cubed and 27 is 3 cubed.
Next, we simplify the square root, √100/81, which equals √(10/9)². This simplifies to 10/9.
Finally, we multiply (2/3) by (10/9) to get the simplified expression, which equals 20/27.
Therefore, the simplified result of the expression (16/81)³/⁴ × √100/81 is 20/27.