Final answer:
The expression 3 times the square root of 8 raised to the power of 1/4 times x is equivalent to 3 times 8 to the 1/8 power times x. This simplification uses rules of exponents and roots specifying that a square root can be expressed as raising to the power of 0.5.
Step-by-step explanation:
The student's question, 'which is equivalent to 3 square root 81/4x?' involves simplifying an expression with a square root and an exponent, using rules of exponents and roots.
To find an equivalent expression, we can apply the property that the square root of a number is the same as that number raised to the power of 0.5. We can also use the fact that the fourth root (or any root) of a number is the same as that number raised to the reciprocal of that root's number (1/4 in this case).
So, 3sqrt(81/4x) can be re-written as 3 * 81/4 * 1/2x = 3 * 81/8x.
Let's consider another example similar to the student's question: if we had (2x)2 = 4.0 (1 − x)2, we could take the square root of both sides and arrive at the simplified expression (2x)(1 − x). This process demonstrates how to handle square roots and fourth roots as fractional exponents.