Question 397063
Is it
{{{(root(3, 4x+1))^6}}}
or
{{{root(3, (4x+1)^6)}}}?
In this particular expression it ends up not making any difference. But in other situations it may make a big difference. So please use parentheses to make things clear. Use
"(the cubed root of 4x+1) to the 6th power" for the first one
and
"the cubed root of (4x+1 to the 6th power)" for the second one. Tutors are more likely to help if the problem is clear.<br>
A quick way to simplify this expression takes advantage of fractional exponents. The general pattern for the relationship between radical expressions and expressions using fractional exponents is:
{{{root(a, q^b) = (root(a, q))^b = x^(b/a)}}}
As you can see, both {{{(root(3, 4x+1))^6}}} and {{{root(3, (4x+1)^6)}}} are both equal to:
{{{(4x+1)^(6/3)}}}
The exponent simplifies to a 2:
{{{(4x+1)^2}}}
And last of all we can use FOIL or the {{{(a+b)^2 = a^2 +2ab + b^2}}} pattern to square 4x+1. (I like using the pattern.)
{{{(4x)^2 + 2(4x)(1) + (1)^2}}}
which simplifies to:
{{{16x^2 +8x + 1}}}
This is the simplified result for both {{{(root(3, 4x+1))^6}}} and {{{root(3, (4x+1)^6)}}}.