Question

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

Answer 1

`200x^(13)y^(18)`

Explanation:

`"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)`

Answer 2

See a solution process below:

Explanation:

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`