Question 76330
Problem #1

If we have

{{{-root(4,16)}}}
Then we are asking: what number multiplied by itself 4 times equals 16? In other words:

{{{x*x*x*x=x^4=16}}}

So if we take the negative fourth root of 16 with a calculator, we get 

{{{-root(4,16)=-2}}}

Since -2 can be represented by a fraction (ie {{{-2=-2/1}}}) it is a rational number.


Problem #2

{{{sqrt (x^4y^3)}}}
{{{sqrt (x^4)*sqrt(y^3)}}} Rewrite the square root of a product as a product of square roots (ie {{{sqrt(xy)=sqrt(x)sqrt(y)}}})
{{{(x^4)^(1/2)*(y^3)^(1/2)}}}Since any root can be written as a fractional power (ie {{{root(n,x)=x^(1/n)}}} rewrite the square roots as exponents of 1/2.
{{{(x^(4*(1/2)))*(y^(3*(1/2)))}}} Multiply the exponents
{{{(x^2)*(y^((3/2)))}}} So this is your simplified answer. It can also be written as:
{{{(x^2)*sqrt(y^3)))}}}