How do you simplify #(5x^2y^3)^2*(2x^3y^4)^3# and write it using only positive exponents?

#"by appling the "color(blue)"laws of exponents"#

#•color(white)(x)(a^m)^(n)=a^((mxxn))larr(color(red)(1))#

#•color(white)(x)a^mxxa^n=a^((m+n))#

#(color(red)(1))" is extended to include all factors inside the parenthesis"#

#rArr(5x^2y^3)^2=5^((1xx2))xx x^((2xx2))xxy^((3xx2))#

#color(white)(xxxxxxxx)=5^2xx x^4xx y^6=25x^4y^6#

#rArr(2x^3y^4)^3=2^((1xx3))xxx^((3xx3))xxy^((4xx3))#

#color(white)(xxxxxxxx)=2^3xx x^9xxy^(12)=8x^9y^(12)#

#rArr(5x^2y^3)^2xx(2x^3y^4)^3#

#=25x^4y^6xx8x^9y^(12)#

#=(25xx8)xx(x^4xx x^9)xx(y^6xxy^(12))#

#=200xx x^((4+9))xxy^((6+12))#

#=200x^(13)y^(18)#

First, use these rules for exponents to eliminate the outer exponents for each term:

or #a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#

#(5x^2y^3)^2 * (2x^3y^4)^3 =>#

#(5^color(red)(1)x^color(red)(2)y^color(red)(3))^color(blue)(2) * (2^color(red)(1)x^color(red)(3)y^color(red)(4))^color(blue)(3) =>#

#(5^(color(red)(1) xx color(blue)(2))x^(color(red)(2) xx color(blue)(2))y^(color(red)(3) xx color(blue)(2))) * (2^(color(red)(1) xx color(blue)(3))x^(color(red)(3) xx color(blue)(3))y^(color(red)(4) xx color(blue)(3))) =>#

#(5^2x^4y^6) * (2^3x^9y^12) =>#

#(25x^4y^6) * (8x^9y^12)#

Next, rewrite the expression as:

#(25 * 8)(x^4 * x^9)(y^6 * y^12) =>#

#200(x^4 * x^9)(y^6 * y^12)#

Now, use this rule of exponents to complete the simplification:

#x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#

#200(x^color(red)(4) * x^color(blue)(9))(y^color(red)(6) * y^color(blue)(12)) =>#

#200x^(color(red)(4)+color(blue)(9))y^(color(red)(6)+color(blue)(12)) =>#

#200x^13y^18#