Radical expressions and equations
Radical Expressions and Equations
Radical Expressions and Equations is the opening topic of Radical Functions in the California 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 \(a,b\).