Separate perfect squares
Factor the radicand into a perfect square times what remains, then take the square root of the perfect-square factor.
Example√72 = √(36 · 2) = 6√2.
Chapter 10 of 10
Radicals
How do roots undo powers, and how can radical expressions be simplified?
A square root asks which nonnegative number squares to produce the radicand. Radical expressions can often be simplified by separating perfect-square factors. Their operations follow the same structural rules as other algebraic expressions.
The big ideas
Understand these before you worry about speed.
Combine like radicals
Radicals add or subtract only when their simplified radicands match.
Example3√5 + 2√5 = 5√5, but √3 + √5 cannot combine.
Solve radical equations by isolating the root
First isolate the radical expression. Square both sides to remove a square root, solve the resulting equation, and then substitute every candidate into the original equation. Squaring is not reversible in every situation, so the final check is required.
Example gives , then . Checking: .
Check for extraneous solutions
Squaring both sides can create a value that does not satisfy the original radical equation. Substitute every answer back.
ExampleFor √(x + 1) = x − 1, any solution must also have x ≥ 1.
Worked example
Follow the reason for each step—not just the symbols.
Problem
Simplify 2√48 − √27.
- Simplify √48 = √(16 · 3) = 4√3, so 2√48 = 8√3.
- Simplify √27 = √(9 · 3) = 3√3.
- Subtract like radicals: 8√3 − 3√3 = 5√3.
Answer5√3
Try it yourself
Work on paper first. Open each answer only when you are ready to check.
1
Simplify √50.
Check answer
5√22
Simplify 4√7 − √7.
Check answer
3√73
Multiply √6 · √15 and simplify.
Check answer
3√104
Solve √(x + 5) = 4.
Check answer
x = 11Practice and tools
Use the resource that matches what you need next.
Worksheet
Square Roots
Reinforce perfect squares and root evaluation.
Open resource WorksheetRadical Expressions
Simplify, combine, multiply, and rationalize radicals.
Open resource WorksheetQuadratic Formula Practice
Apply radical simplification when solving quadratics.
Open resource QuizAlgebra 1 Final Test
Answer 20 questions covering all ten chapters, then review every solution.
Open resource