Simplify with only positive exponents:
(v^-1/2t)^-2(t^2/v^-2)^-3
To simplify the expression with positive exponents, we'll follow these steps:
Step 1: Start by applying the power of a power rule. When raising a power to another power, multiply the exponents.
(v^(-1/2t))^(-2) = v^((-1/2t)*(-2)) = v^t
Step 2: Next, simplify the second part of the expression:
(t^2/v^(-2))^(-3) = (t^2 * v^(-2))^(-3)
Step 3: Now, let's simplify the term inside the parentheses. When multiplying two expressions with the same base, we add the exponents.
t^2 * v^(-2) = t^2 * (1/v^2) = t^2/v^2
Step 4: Apply the power of a product rule. When raising a quotient to another power, apply the power to the numerator and denominator separately.
(t^2/v^2)^(-3) = (t^2)^(-3) / (v^2)^(-3)
Step 5: Simplify further by applying the power rule for exponents:
(t^2)^(-3) = t^(-6)
(v^2)^(-3) = (1/v^2)^3 = 1/v^6, since raising a reciprocal to a power results in flipping the fraction and applying the power to both the numerator and denominator.
Step 6: Combine the simplified terms:
(t^2/v^2)^(-3) = t^(-6) / (1/v^6) = t^(-6) * v^6 = v^6 / t^6
Therefore, the simplified expression with only positive exponents is v^t * v^6 / t^6, or written as (v^t * v^6) / t^6.