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Solving Problems with Radicals
Roots are written using a radical sign, and a number denoting which root to solve for. When none is given, it is an implied square root.
Learning Objective

Solve radical equations by using the squaring property of equality
Key Points
 Roots are usually written using the radical symbol, but can also be written by raising the number to a fraction. Then, the root is the inverse of the raised power. Like this:
$\sqrt x = x^{\frac12}$ .  To solve an equation with a radical: isolate the radical on one side of the equation, get rid of your radical, solve the remaining equation.
 To eliminate a square root, square the radical, to eliminate a cubed root, cube the radical  don't forget to do the exact same thing to the other side of the equation!
Term

radical
A root (of a number or quantity).
Full Text
Roots are written using a radical sign. If there is no denotation, it is implied that you are finding the square root. Otherwise, a number will appear denoting which root to solve for. Any expression containing a radical is called a radical expression.
The best way to solve an equation, is to start by simplifying it as much as possible. You want to start by getting rid of the radical. Do this by treating the radical as if it where a variable. Isolate it on one side and go from there.
Let's look at how to do it stepbystep:
1. Isolate the radical on one side of the equation.
2. Get rid of your radical (some of the rules listed below may help in this).
3. Repeat steps 1&2 if you have another radical.
4. Solve the remaining equation.
5. Double check equation by plugging in your answer.
And remember, always treat each side of the equation the same, here's some helpful reminders for general equation solving: .
Properties of Equality
Make sure to use these properties when solving an equation.
Some helpful rules:
1.
2.
3.
4.
5.
Let's run through an example:
Solve the following:
1. Isolate the radical (already done).
2. Get rid of the radical:
No more radicals? Great, solve for x:
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Key Term Reference
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 root
 Appears in these related concepts: Introduction to Radicals, The Rule of Signs, and Zeroes of Polynomial Functions with Real Coefficients
 sign
 Appears in these related concepts: Polynomial Inequalities, The Intermediate Value Theorem, and Sequences of Mathematical Statements
 square
 Appears in these related concepts: Matrix Multiplication, Factoring a Difference of Squares, and Radical Equations
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
Sources
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Cite This Source
Source: Boundless. “Solving Problems with Radicals.” Algebra. Boundless, 20 Sep. 2016. Retrieved 01 Oct. 2016 from https://www.boundless.com/users/6970/textbooks/algebra18383e3468f14c72a528a0916137c596/thebuildingblocksofalgebra1/radicalnotationandexponents14/solvingproblemswithradicals815897/