How do you simplify #(root4(x^3)*root4(x^5))^(-2)#?

1 Answer
Aug 28, 2017

Answer:

See a solution process below:

Explanation:

First, we can use this rule to combine the radicals within the parenthesis:

#root(n)(color(red)(a)) * root(n)(color(blue)(b)) = root(n)(color(red)(a) * color(blue)(b))#

#(root(4)(color(red)(x^3)) * root(4)(color(blue)(x^5)))^-2 = (root(4)(color(red)(x^3) * color(blue)(x^5)))^-2#

Next, use this rule for exponents to combine the terms within the radical:

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

#(root(4)(color(red)(x^3) * color(blue)(x^5)))^-2 = (root(4)(x^(color(red)(3)+color(blue)(5))))^-2 = (root(4)(x^8))^-2#

Then, we can use this rule to rewrite the radical into an exponent:

#root(color(red)(n))(x) = x^(1/color(red)(n))#

#(root(color(red)(4))(x^8))^-2 = ((x^8)^(1/color(red)(4)))^-2#

Next, we can use this rule to simplify the inner exponents:

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

#((x^color(red)(8))^color(blue)(1/4))^-2 = (x^(color(red)(8) xx color(blue)(1/4)))^-2 = (x^2)^-2#

We can use the same rule to reduce the outer exponents:

#(x^color(red)(2))^color(blue)(-2) = x^(color(red)(2) xx color(blue)(-2)) = x^-4#

We can now use this rule to eliminate the negative exponent:

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

#x^color(red)(-4) = 1/x^color(red)(- -4) = 1/x^4#