How do you simplify #-root8(4)+5root8(4)#?

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Answer 1 · 2017-06-12T14:26:22

See a solution process below:

First, rewrite this expression as:

#-1root(8)(4) + 5root(8)(4)#

Next, factor and combine these like terms:

#(-1 + 5)root(8)(4) = 4root(8)(4)#

We can now rewrite this expression using this rule for exponents and radicals:

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

#4root(color(red)(8))(4) = 4 * 4^(1/color(red)(8))#

We can now use these two rules of exponents to again rewrite this expression:

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

#4 * 4^(1/8) => 4^color(red)(1) xx 4^color(blue)(1/8) = 4^(color(red)(1) + color(blue)(1/8)) = 4^(color(red)(8/8) + color(blue)(1/8)) = 4^(9/8)#

Or, depending on the solution you are looking for, using the reverse of the rule above:

#4^(9/8) = 4^(9 * 1/8) = (4^9)^(1/8) => root(8)(4^9)#

Some of the simplifications are, depending on what you are working on:

#4root(8)(4)# or #4^(9/8)# or #root(8)(4^9)#