How do you simplify the expression #((x^7y^-2)/(3y^-3))^-2# using the properties?

1 Answer
Jun 15, 2017

Answer:

See a solution process below:

Explanation:

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

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

#((x^7y^-2)/(3y^-3))^-2 => ((x^color(red)(7)y^color(red)(-2))/(3^color(red)(1)y^color(red)(-3)))^color(blue)(-2) => (x^(color(red)(7)*color(blue)(-2))y^(color(red)(-2)*color(blue)(-2)))/(3^(color(red)(1)*color(blue)(-2))y^(color(red)(-3)*color(blue)(-2))) =>#

#(x^-14y^4)/(3^-2y^6)#

Next, use this rule of exponents to simplify the #y# terms:

#x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))#

#(x^-14y^color(red)(4))/(3^-2y^color(blue)(6)) => x^-14/(3^-2y^(color(blue)(6)-color(red)(4))) => x^-14/(3^-2y^2)#

Then use this rule of exponents to eliminate the negative exponent for the #x# term:

#x^color(red)(a) = 1/x^color(red)(-a)#

#x^color(red)(-14)/(3^-2y^2) => 1/(3^-2x^color(red)(- -14)y^2) => 1/(3^-2x^14y^2)#

Now, use this rule of exponents to eliminate the negative exponent for the #3# term:

#1/x^color(red)(a) = x^color(red)(-a)#

#1/(3^color(red)(-2)x^14y^2) => 3^color(red)(- -2)/(x^14y^2) => 3^2/(x^14y^2) => 9/(x^14y^2)#