A number under a radical can often be tidied into a simpler form.
Since √12 is 4 times 3, the square factor 4 steps out as a 2, giving a neat 2√3.
Radicals with the same inside add by their coefficients, products and quotients act inside the root, and a root in the denominator is cleared by rationalizing.
Here you slide to change the number, watch factors pair up and step out, and see like radicals merge as bars.
Simplifying Radicals — Use Prime Factorization
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Simplifying √n via prime factorization
12
🔧 Simplification Principle
①√(a²×b) = a√b — paired factors come out
②e.g. √12 = √(4×3) = 2√3
③e.g. √48 = √(16×3) = 4√3
Adding · Subtracting — Like Radicals Only!
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Adding the coefficients of like radicals
3
2
📏 Key to Addition
①Same radical part needed to combine (like terms)
②2√3 + 5√3 = 7√3 ✓
③2√3 + 5√2 stays as-is ✗
④Simplify first to like radicals: √12 + √27 = 2√3 + 3√3 = 5√3
Multiplying & Dividing Radicals
Product Rule
√a × √b = √(ab)
Radicals multiply → multiply inside
Quotient Rule
√a√b = √(ab)
Radicals divide → divide inside
💡 Mul/Div Points
①√2 × √3 = √6
②√6 ÷ √2 = √3
③(2√3)² = 4 × 3 = 12 — coefficient also squared!
Rationalizing the Denominator
Rationalize
a√b = a√bb
Multiply top & bottom by √b to remove the radical
🧹 Why Rationalize
①√ in denominator makes comparison hard
②Multiplying both by the same √ makes the bottom rational
③e.g. 1/√2 = √2/2 ≈ 0.707...
Work It Out
Example 1
Compute √2 × √8.
1
A product of radicals combines as √a × √b = √(ab).
√2 × √8 = √(2 × 8) = √16
2
√16 = 4.
= 4
▸ 4
Multiplying radicals multiplies the radicands.
Example 2
Compute 2√3 + 5√3.
1
Like radical terms add by their coefficients.
(2 + 5)√3
2
Compute.
= 7√3
▸ 7√3
Only matching radical parts can be added like terms.