grade 9 Operations with Radicals

Operations with Radicals

A number under a radical can often be tidied into a simpler form. Because √12 is 4 times 3, the square factor 4 steps out as a 2, giving 2√3. Radicals with the same inside, like 2√3 and 5√3, add by their coefficients; products and quotients act inside the root; a root in the denominator is cleared by rationalizing. The sliders move simplification and like-radical addition; products, quotients, and rationalizing stay in the written boxes below.

Simplifying Radicals — Use Prime Factorization

n value12

🔧 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!

a (front coeff)3

b (back coeff)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 ✗. The bars below add like √2 terms only

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!

④√a + √b ≠ √(a+b)

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.

A product of radicals combines as √a × √b = √(ab).

√2 × √8 = √(2 × 8) = √16

√16 = 4.

= 4

▸ 4

Multiplying radicals multiplies the radicands.

Example 2

Compute 2√3 + 5√3.

Like radical terms add by their coefficients.

(2 + 5)√3

Compute.

= 7√3

▸ 7√3

Only matching radical parts can be added like terms.

Exam Wrap-up

Radical Operations Recap

a√m ± b√m = (a±b)√m

Like radicals → operate on coefficients

Grade-9 school exam type

Rationalize the denominator of 6/√3.

①2√3

②3√3

③√3

④6√3

⑤2

▸ ① 2√3

Multiply numerator and denominator by √3.

6/√3 × √3/√3 = 6√3/3

Reduce.

= 2√3

🎯 Exam Points

①Simplify first! √48 + √12 → 4√3 + 2√3 = 6√3

②Don't forget to rationalize

③Use (√a)² = a in calculations

④√a × √b = √(ab) but √a + √b ≠ √(a+b)

⑤Inside a radical can't be negative

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Polynomial Multiplication & Formulas

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