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

1 Answer
Jun 15, 2017

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)