Write square root q^3 ^3 square root q^8 as a single radical using the smallest possible root
To simplify the expression, we can rewrite it as a single radical using the smallest possible root.
First, let's simplify the exponents using the property (a^m)^n = a^(m*n):
(square root(q^3))^3 * square root(q^8)
Now, taking the cube of the square root of q^3:
(squareroot(q^3))^3 = (q^3)^(1/2 * 3) = q^(3/2 * 3) = q^(9/2)
And, taking the square root of q^8:
square root(q^8) = (q^8)^(1/2) = q^(8/2) = q^4
Finally, the expression can be written as:
q^(9/2) * q^4
Combining the terms with the same base:
q^(9/2 + 4) = q^(9/2 + 8/2) = q^(17/2)
So, the simplified expression is square root q^3 ^3 square root q^8 = q^(17/2).