# Algebra Examples

Prove that a Root is on the Interval
,
The Intermediate Value Theorem states that, if is a real-valued continuous function on the interval , and is a number between and , then there is a contained in the interval such that .
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Raise to the power of .
Calculate .
Remove parentheses around .
Raise to the power of .

Rewrite the equation as .
Take the cube root of both sides of the equation to eliminate the exponent on the left side.
Simplify the right side.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
The Intermediate Value Theorem states that there is a root on the interval because is a continuous function on .
The roots on the interval are located at .