Before we attempt to simplify the given radical expression, let's consider one of the more difficult parts of correctly simplifying radicals. When the exponent of a variable inside the radical is even and the simplified expression for that variable has an odd exponent, we need to use absolute value symbols.
ll
sqrt(x^2)= |x| & sqrt(x^3)=x sqrt(x)
sqrt(x^4)= x^2 & sqrt(x^6)=|x^3|
This is because we do not want the result to be negative — the range of a square root function is all real numbers greater than or equal to 0. Now, consider the given radical expression.
sqrt(2ab^2)* sqrt(10a^5b)
The exponent of b is even and, in the simplified expression, this variable will have an odd exponent. However, since in the second factor b has no exponent, it must be positive for the expression to make sense. Furthermore, the exponent of a is odd. Therefore, neither variable will need an absolute value symbol when removed from the radical.
Now that we have simplified the factors as much as possible, let's recall the Product Property of Square Roots.
sqrt(a* b)=sqrt(a) * sqrt(b), for a≥ 0,b≥ 0
Let's use this property to simplify our expression even further.
