Algebra 2 · Grades 10, 11

Simplifying Radical Expressions

Quick answer

A radical is in simplest form when no perfect-power factor remains under it, no fraction sits under it, and no radical is left in a denominator. The product rule, the root of ab equals the root of a times the root of b, pulls perfect squares out of a square root: √72 = √36 · √2 = 6√2. Like radicals combine like terms, as in 3√2 + 5√2 = 8√2. A radical in a denominator is cleared by multiplying by the radical itself or, for a binomial, by its conjugate: 1/(√3 + 1) = (√3 − 1)/2.

What you'll learn

  • Simplify square roots and cube roots by removing perfect powers
  • Multiply, divide, add and subtract radical expressions
  • Rationalize denominators, including with conjugates
  • Simplify radicals that contain variables

Taking out perfect powers

A square root is simplest when nothing under it is a perfect square beyond 11. The key is the product rule:

ab=a bab3=a3 b3\sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad\qquad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}

Split off the largest perfect square, take its root, and leave the rest:

72=36⋅2=62543=27⋅23=323\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \qquad\qquad \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}

For a cube root the factor to pull out is a perfect cube.

Why the product rule holds

a b\sqrt{a}\,\sqrt{b} is a positive number whose square is (a)2(b)2=ab\left(\sqrt{a}\right)^2\left(\sqrt{b}\right)^2 = ab, and the positive number whose square is abab is ab\sqrt{ab}. So the two are equal. Rationalizing a binomial denominator uses a second product,

(a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2

which squares both terms: multiplied by its conjugate, a sum or difference with a square root in it loses the root. Radicals are simplified with two multiplications, the product rule and the difference of squares, and every step can be checked by squaring or multiplying back.

Multiplying and dividing

Multiply radicals with the product rule, and expand binomials the usual way:

(2+3)(2−3)=4−3=1(5+2)2=5+45+4=9+45(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1 \qquad\qquad \left(\sqrt{5} + 2\right)^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5}

Rationalizing the denominator

A simplified answer has no radical in its denominator. Multiply top and bottom by the radical, or, for a binomial, by its conjugate:

63=633=2343−5=4(3+5)9−5=3+5\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} \qquad\qquad \frac{4}{3 - \sqrt{5}} = \frac{4\left(3 + \sqrt{5}\right)}{9 - 5} = 3 + \sqrt{5}

Worked examples

Common mistakes

Practice problems

  1. Simplify 48\sqrt{48}.

    Answer

    434\sqrt{3}

    Full solution

    48=16⋅348 = 16 \cdot 3, and 16=4\sqrt{16} = 4.

  2. Simplify 403\sqrt[3]{40}.

    Answer

    2532\sqrt[3]{5}

    Full solution

    40=8⋅540 = 8 \cdot 5, and 83=2\sqrt[3]{8} = 2.

  3. Simplify 12+27\sqrt{12} + \sqrt{27}.

    Answer

    535\sqrt{3}

    Full solution

    23+332\sqrt{3} + 3\sqrt{3}.

  4. Simplify 6⋅15\sqrt{6} \cdot \sqrt{15}.

    Answer

    3103\sqrt{10}

    Full solution

    90=9⋅10\sqrt{90} = \sqrt{9 \cdot 10}.

  5. Rationalize 105\displaystyle\frac{10}{\sqrt{5}}.

    Answer

    252\sqrt{5}

    Full solution

    1055=25\tfrac{10\sqrt{5}}{5} = 2\sqrt{5}.

  6. Rationalize 32−1\displaystyle\frac{3}{\sqrt{2} - 1}.

    Answer

    32+33\sqrt{2} + 3

    Full solution

    Multiply by the conjugate 2+1\sqrt{2} + 1: the denominator becomes 2−1=12 - 1 = 1.

  7. Simplify 81x3\sqrt{81x^3} for x≥0x \ge 0.

    Answer

    9xx9x\sqrt{x}

    Full solution

    81x3=81x2⋅x81x^3 = 81x^2 \cdot x, and 81x2=9x\sqrt{81x^2} = 9x for x≥0x \ge 0.

  8. Expand (3−2)2\left(3 - \sqrt{2}\right)^2.

    Answer

    11−6211 - 6\sqrt{2}

    Full solution

    9−62+29 - 6\sqrt{2} + 2.

  9. Simplify (7+3)(7−3)\left(\sqrt{7} + \sqrt{3}\right)\left(\sqrt{7} - \sqrt{3}\right).

    Answer

    44

    Full solution

    A difference of squares: 7−37 - 3.

  10. A student writes x2+9=x+3\sqrt{x^2 + 9} = x + 3. What went wrong?

    Hint

    Test x=4x = 4.

    Answer

    A root does not split over a sum. x2+9\sqrt{x^2 + 9} does not simplify.

    Full solution

    At x=4x = 4: 16+9=5\sqrt{16 + 9} = 5, but 4+3=74 + 3 = 7. Squaring the student’s answer gives x2+6x+9x^2 + 6x + 9, not x2+9x^2 + 9.

Frequently asked questions

How do you simplify a square root?

Write the number as a perfect square times another factor and take the square root of the perfect square: √72 = √(36 · 2) = 6√2.

When can radicals be added?

Only when they are like radicals, with the same index and the same number under the root: 3√2 + 5√2 = 8√2. √2 + √3 cannot be combined.

What does rationalizing the denominator mean?

Rewriting a fraction so that no radical is left in the denominator, by multiplying top and bottom by a suitable radical or conjugate.

What is a conjugate?

The same binomial with the middle sign changed: the conjugate of 3 − √5 is 3 + √5. Their product, 9 − 5 = 4, has no radical.

Is √(a + b) equal to √a + √b?

No. √(9 + 16) = √25 = 5, but √9 + √16 = 7. The root of a product splits; the root of a sum does not.

What to learn next

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.HSN.RN.A.2The Real Number SystemRewrite expressions involving radicals and rational exponents using the properties of exponents.
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