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.

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:<equation contenteditable="false">$\sqrt x = x^{\frac12}$.

• 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!

Terms

• A root (of a number or quantity).

Figures

1. Properties of Equality

Make sure to use these properties when solving an equation.

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 step-by-step:

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: Figure 1.

1. $\sqrt {x} \cdot \sqrt {x} = {(\sqrt {x})}^{2}=x$

2. $\sqrt{\frac{x}{y}}= \frac {\sqrt{x}}{\sqrt{y}}$

3. ${x}^{\frac{m}{n}} = {( \sqrt[n]{x})}^{m}$

4.  $\sqrt {x} \sqrt {y}=\sqrt {xy}$

5.  $\sqrt [m]{\sqrt [n]{x}}= \sqrt [m \cdot n]{x} = {x}^{\frac{1}{m \cdot n}}$

Let's run through an example:

Solve the following: $\sqrt {2x+5} = 7$

2. Get rid of the radical: ${(\sqrt {2x+5})}^{2}={(7)}^{2}$

$2x+5 = 49$

No more radicals?  Great, solve for x:

$2x+5-5=49-5$

$2x=44$

$\frac{2x}{2}=\frac{44}{2}$

$x=22$

Key Term Glossary

equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
Appears in these related concepts:
expression
An arrangement of symbols denoting values, operations performed on them, and grouping symbols. E.g. (2x+4)/2
Appears in these related concepts:
A root (of a number or quantity).
Appears in these related concepts:
root
the number which,when  plugged into the equation, will produce a zero.
Appears in these related concepts:
sign
positive or negative polarity.
Appears in these related concepts:
square
The second power of a number, value, term or expression.
Appears in these related concepts:
variable
A symbol that represents a quantity in a mathematical expression, as used in many sciences