Question 383778
{{{root(4, 243x^8y^10)}}}
I'm not sure I can figure out how you got the answer you did. So I will not be able to explain what went wrong. But your Math book is correct.<br>
Your expression is a 4th root. And to simplify a 4th root you look for factors of the radicand (the expression within the radical) that are 4th powers. (Other factors are of no real interest.)<br>
For 243 there may be many ways to factor it but one way, 81*3, involves a factor that is a 4th power. ({{{81 = 3^4}}}) For the variables we factor each one into as many powers of 4 as we can. Putting all these factors together we get:
{{{root(4, 3^4*3*x^4*x^4*y^4*y^4*y^2)}}}
Take a moment to make sure you see how your original radicand and the factored one you see above are equal to each other.<br>
Next, I like to use the Commutative Property of Multiplication to reorder the factors so that all the 4th powers are in front:
{{{root(4, 3^4*x^4*x^4*y^4*y^4*3*y^2)}}}
Next we use a property of radicals, {{{root(a, p*q) = root(a, p)*root(a, q)}}} to separate all the 4th power factors into their own radicals. (The factors that are not 4th powers (at the end) just stay all in the same radical.):
{{{root(4, 3^4)*root(4, x^4)*root(4, x^4)*root(4, y^4)*root(4, y^4)*root(4, 3*y^2)}}}
Now all the 4th roots of the 4th powers simplify:
{{{3*x*x*y*y*root(4, 3y^2)}}}
which simplifies to:
{{{3*x^2*y^2*root(4, 3y^2)}}}
which is the answer in your book.