Solving radicals and rational exponents requires creating equations that are different from the original equations. This may result in getting extraneous solutions, meaning that the solutions to the new equations don’t satisfy the original equations and therefore are not correct.
Rational expressions:
A rational expression is an expression containing at least one fraction with a variable in the denominator. The variables in the nominator and the denominator can be a quadratic or a higher order polynomial. For example: 3x/(x2-2) is a rational expression with linear variable in the nominator and a quadratic variable in the denominator.
Solving rational expressions includes simplifying, performing 4 operations with fractions (adding, subtracting, dividing, and multiplying) and cancelling common factors.
Rewriting a rational expression as a quotient and a remainder can be done with 2 methods: long division and grouping the numerator.
Rational equations:
A rational equation is an equation containing at least one fraction with a variable in the denominator. For example: 2/(x+2)=1.
Solving rational equations is done by multiplying both sides of the equation by the least common denominator. Since rational expressions contain a variable in the denominator, we need to exclude an extraneous solution for which the denominator equals to zero (we can’t divide by 0).
Radical equations:
Radicals are rational exponents that are written with roots. For example √x. The symbol of a radical is √ and it represents a square root (instead of writing 2√x we write only √x).
A radical equation is an equation in which a variable appears under a radical. For example: √(x+1)=1 is a radical equation and √(4+12)=x is not a radical equation.
Solving radical equation is done by squaring both of its sides, this action cancels the radical sign and results in a linear or quadratic equation that we can solve. Extraneous solution is a solution that we get after solving a squared equation that is not a solution to the original equation. We need to exclude extraneous solutions since they are incorrect.
Previous knowledge for this topic includes solving quadratic equations, operations with exponents and solving a linear equation with fractions topics.
Continue reading this page for detailed explanations and examples.