Radical Expressions
Radicals are roots written with the √ symbol.
About this topic
A radical expression contains a root symbol, such as √12 or ∛x. To simplify, factor out perfect squares (or cubes): √12 = √(4 · 3) = 2√3. Multiply radicals by multiplying under the root, and add them only when the radicals match.
To remove a radical from a denominator, multiply top and bottom by a suitable expression. For 1/√3 multiply by √3/√3 to get √3/3. When the denominator is a binomial like 2 + √5, multiply by its conjugate 2 − √5, using the difference of squares to eliminate the root. Like radicals combine just like like terms: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined.
Radicals appear in the Pythagorean theorem and the distance formula, in the quadratic formula, and in physics formulas for pendulum periods and wave speeds. Simplified form is the standard way to give exact answers.
The classic errors are claiming that √(a + b) equals √a + √b, which is false (√(9 + 16) is 5, not 7), and forgetting to simplify fully. Also note that the square root symbol means the positive root only.
Key ideas
- √(ab) = √a × √b
- Simplify by pulling out perfect squares
- Rationalize denominators
Quick check
Use cases: where Radical Expressions helps
- Homework: review the key idea behind Radical Expressions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Radical Expressions without notes.
- Visual learning: play with the interactive lab to see how Radical Expressions changes when you adjust its inputs.
- Teaching a friend: explain Radical Expressions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Radical Expressions?
Radicals are roots written with the √ symbol.
What will I learn in Radical Expressions?
√(ab) = √a × √b. Simplify by pulling out perfect squares. Rationalize denominators.
Is Radical Expressions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Radical Expressions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Radical Expressions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.