How do you simplify #((2p m^-1q^0)^-4*2m^-1p^3)/(2pq^2)# and write it using only positive exponents?

Question

Hazel Anglin · Accepted Answer

See a solution process below:

First, rewrite the expression as: #((2 * 2)/2)((p * p^3)/p)(m^-1 * m^-1)(q^0/q^2)# Next, cancel common terms in the numerator and denominator: #((color(red)(cancel(color(black)(2))) * 2)/color(red)(cancel(color(black)(2))))((color(blue)(cancel(color(black)(p))) * p^3)/color(blue)(cancel(color(black)(p))))(m^-1 * m^-1)(q^0/q^2) =># #2p^3(m^-1 * m^-1)(q^0/q^2)# Next, use this rule of exponents to simplify the #q# term: #a^color(red)(0) = 1# #2p^3(m^-1 * m^-1)(q^color(red)(0)/q^2) =># #2p^3(m^-1 * m^-1)(1/q^2) =># #(2p^3)/(q^2)(m^-1 * m^-1)# Now, use these rules of exponents to simplify the #m# terms: #x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))# and #x^color(red)(a) = 1/x^color(red)(-a)# #(2p^3)/(q^2)(m^color(red)(-1) * m^color(blue)(-1)) =># #(2p^3)/(q^2)m^(color(red)(-1)+color(blue)(-1)) =># #(2p^3)/(q^2)m^(color(red)(-2) =># #(2p^3)/(q^2)(1/m^(color(red)(- -2) =># #(2p^3)/(q^2)(1/m^2) =># #(2p^3)/(q^2m^2)#

Jace Anderson · Answer

undefined To simplify the expression and rewrite it using only positive exponents, follow these steps:

1. Expand the expression inside the parentheses:
   ((2p m^-1q^0)^-4 * 2m^-1p^3) / (2pq^2)
   = ((2^-4 * p^-4 * m^4 * q^0) * (2m^-1 * p^3)) / (2pq^2)

2. Combine like terms:
   = (2^-4 * 2 * p^-4 * m^3 * q^0 * p^3) / (2pq^2)

3. Simplify the constants:
   = (2^-3 * p^-4 * m^3 * p^3) / (pq^2)

4. Combine the like terms in the numerator:
   = (2^-3 * p^-1 * m^3) / (pq^2)

5. Rewrite the expression with positive exponents:
   = (1 / (2^3 * p * m^3)) * (1 / (pq^2))
   = (1 / (8pm^3)) * (1 / (pq^2))

Therefore, the simplified expression with only positive exponents is (1 / (8pm^3)) * (1 / (pq^2)).