Radical Expressions

Grade 9 · Math · Study Guide

Simplify, add, subtract, multiply, and rationalize radical expressions with confidence.

A radical expression contains a root symbol (√). The most common is the square root, but cube roots and higher-order roots also appear. The expression under the radical sign is called the radicand, and the small number indicating the root is the index.

To simplify a square root, factor the radicand and pull out any perfect square factors. For example, √72 = √(36 · 2) = 6√2. A radical is fully simplified when no perfect square factors remain under the sign and no fractions appear under the radical.

Like radicals have the same radicand and index and can be added or subtracted just like like terms: 3√5 + 7√5 = 10√5. Unlike radicals cannot be combined unless you simplify them first to create matching radicands.

To multiply radicals with the same index, multiply the coefficients together and multiply the radicands together: 3√2 · 4√3 = 12√6. Use FOIL when multiplying binomials containing radicals.

Rationalizing the denominator means rewriting a fraction so no radical appears in the denominator. For a monomial denominator like 1/√3, multiply by √3/√3. For a binomial denominator like 1/(√3 + 1), multiply by the conjugate (√3 − 1)/(√3 − 1).

Key terms

Radical
An expression involving a root, written with the √ symbol
Radicand
The expression under the radical sign
Index
The small number above the radical indicating which root to take (2 for square root, 3 for cube root, etc.)
Simplified radical form
A radical with no perfect square factors in the radicand and no radicals in the denominator
Like radicals
Radicals with the same index and the same radicand that can be added or subtracted
Conjugate
The expression formed by changing the sign between two terms in a binomial, used to rationalize denominators

Flashcards

Simplify √50
√(25·2) = 5√2
Simplify √200
√(100·2) = 10√2
Simplify ∛8
2, because 2³ = 8
Add: 2√3 + 4√3
6√3
Subtract: 7√5 − 3√5
4√5
Rationalize 3/√7
3√7/7 (multiply top and bottom by √7)
Multiply (1 + √2)(1 − √2)
1 − 2 = −1 (difference of squares with radicals)
Simplify √(x⁶)
x³ (for x ≥ 0)

Questions

  1. Simplify √48.

    1. 4√3
    2. 6√2
    3. 3√4
    4. 2√12
    Show answer

    Answer: 4√3

    Factor 48 = 16 · 3. Pull out the perfect square: √(16 · 3) = √16 · √3 = 4√3.

  2. Simplify 3√2 + 5√2. The answer is __√2.

    Show answer

    Answer: 8

    Like radicals (same radicand √2) are combined by adding coefficients: 3 + 5 = 8, giving 8√2.

  3. Multiply: √3 · √12

    1. √15
    2. 6
    3. 3√4
    4. 4√3
    Show answer

    Answer: 6

    √3 · √12 = √(3 · 12) = √36 = 6.

  4. √25 + √16 = √41

    Show answer

    Answer: False

    √25 + √16 = 5 + 4 = 9. You cannot add radicals by adding radicands; √41 ≈ 6.4, which is not 9.

  5. Rationalize the denominator: 5/√5

    1. √5
    2. 5√5
    3. 1/√5
    4. 5/5
    Show answer

    Answer: √5

    Multiply numerator and denominator by √5: (5 · √5)/(√5 · √5) = 5√5/5 = √5.

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