Radical expressions and equations
Radical Expressions and Equations
Radical Expressions and Equations is the opening topic of Radical Functions in the Common Core State Standards. It is aligned to Standard N-RN.2, which requires students to solve simple radical equations in one variable and identify extraneous solutions.
Radical expressions simplify by pulling out perfect-square factors; radical equations are solved by isolating the radical and squaring.
Theory
A radical expression contains a root. Two product rules simplify them:
To solve a radical equation, isolate the radical and square both sides.
Product and quotient rules:
How to work with radicals
- Factor out perfect squares to simplify.
- To solve, isolate the radical.
- Square both sides.
- Solve and check each answer.
Factor out the largest perfect square.
| \(\sqrt{50}\) | \(=\) | \(\sqrt{25}\cdot\sqrt{2}\) |
| \(=\) | \(5\sqrt{2}\) |
Combine under one root.
| \(\sqrt{8}\cdot\sqrt{2}\) | \(=\) | \(\sqrt{16}\) |
| \(=\) | \(4\) |
Square both sides.
| \(x+3\) | \(=\) | \(25\) |
| \(x\) | \(=\) | \(22\) |
Square, then solve.
| \(2x-1\) | \(=\) | \(9\) |
| \(2x\) | \(=\) | \(10\) |
| \(x\) | \(=\) | \(5\) |
Common pitfalls
Frequently asked questions
How do you simplify \(\sqrt{50}\)?
Factor out the perfect square: \(\sqrt{50}=5\sqrt2\).
How do you solve a radical equation?
Isolate the radical and square both sides, then check.
Why check solutions?
Squaring can introduce extraneous solutions.
Is \(\sqrt{a}\,\sqrt{b}=\sqrt{ab}\)?
Yes for non-negative \(a,b\).