simplify ((mn^3)^2)*(((2^-2)(m^-1)(n^3))^-2)
First, we need to simplify each term separately before evaluating the powers.
Starting with the first term, (mn^3)^2:
(mn^3)^2 = m^2 * (n^3)^2
= m^2 * n^6
Next, moving on to the second term, (((2^-2)(m^-1)(n^3))^-2):
(((2^-2)(m^-1)(n^3))^-2) = (2^-2)^-2 * (m^-1)^-2 * (n^3)^-2
= (2^4) * (m^2) * (n^-6)
= (16m^2)/n^6
Now, combining the simplified terms and evaluating the powers:
((mn^3)^2)*(((2^-2)(m^-1)(n^3))^-2) = (m^2 * n^6) * ((16m^2)/n^6)
= (m^2 * n^6) * (16m^2) / (n^6)
= 16m^4
Therefore, the simplified expression is 16m^4.
To simplify the expression ((mn^3)^2)*(((2^-2)(m^-1)(n^3))^-2), let's break it down step by step:
Step 1: Simplify the expression inside the first set of parentheses.
((mn^3)^2) becomes (m^2n^6).
Step 2: Simplify the expression inside the second set of parentheses.
((2^-2)(m^-1)(n^3))^-2 can be simplified as follows:
2^-2 = 1/2^2 = 1/4 (using the rule that a negative exponent flips the base)
m^-1 = 1/m (using the rule that a negative exponent flips the base)
(n^3)^-2 = 1/n^(3*2) = 1/n^6 (using the rule that exponents multiply when raised to a power)
Combining all three parts, ((2^-2)(m^-1)(n^3))^-2 simplifies to (1/4)(1/m)(1/n^6) or (1/(4mn^6)).
Step 3: Multiply the simplified expressions together.
Now, we can simplify the overall expression:
((m^2n^6))*((1/(4mn^6))).
After multiplying, the m term cancels out, and the n^6 term cancels out, leaving us with:
(m^2)*(1/4) = m^2/4
Therefore, ((mn^3)^2)*(((2^-2)(m^-1)(n^3))^-2) simplifies to m^2/4.