to simplify the expression p1/2 q2p1/4 q1/2 r1/6 and then evaluate it when p=16, q=9, and r=4, taking only positive roots.

Here's the solution to the problem: The problem asks to simplify the expression p^1/2 q^2p^1/4 q^1/2 r^1/6 and then evaluate it when p=16, q=9, and r=4, taking only positive roots. Step 1: Simplify the expression using the laws of exponents. Recall the exponent rules: (a^m)/(a^n) = a^m-n and a^-n = (1)/(a^n). p^1/2 q^2p^1/4 q^1/2 r^1/6 Combine the terms with the same base: For p: p^1/2 - 1/4 = p^2/4 - 1/4 = p^1/4 For q: q^2 - 1/2 = q^4/2 - 1/2 = q^3/2 For r: The r^1/6 term is in the denominator, so it becomes r^-1/6 when moved to the numerator. The simplified expression is: p^1/4 q^3/2 r^-1/6 This can also be written as: p^1/4 q^3/2r^1/6 Step 2: Evaluate the simplified expression using the given values p=16, q=9, and r=4. Substitute these values into the simplified expression: (16)^1/4 (9)^3/2(4)^1/6 Calculate each part: • (16)^1/4: This is the fourth root of 16. Since 2^4 = 16, (16)^1/4 = 2. (Taking the positive root as specified). • (9)^3/2: This can be written as (sqrt(9))^3. Since sqrt(9) = 3 (taking the positive root), we have 3^3 = 27. • (4)^1/6: This can be written as (2^2)^1/6. Using the rule (a^m)^n = a^mn, we get 2^2 × 1/6 = 2^2/6 = 2^1/3. This is the cube root of 2, or [3]2. Now substitute these calculated values back into the expression: (2 × 27)/([3]2) (54)/([3]2) The final evaluated expression is (54)/([3]2).