Rationalizing the Denominator

No root left in the denominator.

A root in the denominator

A fraction such as 1/√2 has a root in its denominator. Its value is fine, but the form is awkward. To find it as a decimal, you would divide 1 by 1.41421…, a decimal that never ends. And to add 1/√2 to another fraction, you would need a common denominator that holds a root.

So a fraction like this is rewritten with a whole number in the denominator. The new denominator is a rational number, so the rewriting is called rationalizing the denominator.

Multiply by 1

Multiplying a fraction by 1 never changes its value, and √2/√2 is 1, because any number other than 0 divided by itself is 1. So multiply both the numerator and the denominator of 1/√2 by √2, which is 1/√2 × √2/√2. The numerator becomes 1 × √2 = √2, and the denominator becomes √2 × √2.

In the denominator, √2 × √2 is exactly 2, because √2 is the number whose square is 2. So the root is gone from the denominator: 1/√2 = √2/2.

The new form is easy to work with. √2/2 is half of √2, which is about 1.414 ÷ 2 = 0.707.

00.10.20.30.40.50.60.70.80.911/√2 = √2/2

1/√2 and √2/2 are the same point on the number line, a little past 0.7. Only the way the number is written has changed.

Then simplify

The same move works with any single root in the denominator. For 6/√3, multiply the numerator and the denominator by √3. The numerator becomes 6 × √3 = 6√3, and the denominator becomes √3 × √3 = 3, so 6/√3 = 6√3/3.

Now the denominator is 3, and 6 ÷ 3 = 2, so 6√3/3 = 2√3. Check it by multiplying back: 2√3 × √3 = 2 × 3 = 6, so 6 ÷ √3 really is 2√3.

For 10/√5, multiply the numerator and the denominator by √5. The denominator becomes √5 × √5 = 5, so 10/√5 = 10√5/5, and 10 ÷ 5 = 2 gives 2√5.

Common mistakes

Multiply the numerator and the denominator by the same root. Multiplying only the denominator of 1/√2 by √2 gives 1/2, which is a different number: 0.5, not 0.707.

Keep the root after dividing. In 6√3/3, the 3 in the denominator divides the 6, and the √3 stays in the numerator: the answer is 2√3, not 2 and not 6√3.

Worked example: The Ratio of a Square's Side to Its Diagonal

Question A diagonal of a square cuts the square into two right-angled triangles, so by Pythagoras' theorem the square of the diagonal is equal to the sum of the squares of two sides. (a) Write the ratio of the side of a square to its diagonal as a fraction with a rational denominator. (b) A square mirror has a diagonal of 60 cm. Find the exact length of one side of the mirror.

  1. 1.Let the side of the square be s and the diagonal be d. By Pythagoras' theorem, d2 = s2 + s2 = 2s2, so d = √2s2 = s√2.

    dssd2= s2+ s2= 2s2, so d = s√2
    dssd2= s2+ s2= 2s2, so d = s√2
    By Pythagoras' theorem, d2 = s2 + s2 = 2s2, so d = s√2.

  2. 2.The ratio of the side to the diagonal is ss√2. Divide the numerator and the denominator by s: the ratio is 1√2.

    s√2ssd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√2
    s√2ssd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√2
    The ratio of the side to the diagonal is ss√2 = 1√2.

  3. 3.(a) Rationalize the denominator: multiply the numerator and the denominator by √2. 1 × √2√2 × √2 = √22, because √2 × √2 = 2.

    s√2ssd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√21/√2= (1 ×√2) / (√2×√2) =√2/2
    s√2ssd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√21/√2= (1 ×√2) / (√2×√2) =√2/2
    (a) Multiply the numerator and the denominator by √2: 1√2 = √22.

  4. 4.The side of any square is √22 of its diagonal. For the mirror, the side is 60 × √22 = 30√2 cm.

    60 cmssd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√21/√2= (1 ×√2) / (√2×√2) =√2/2side = 60 ×√2/2 = 30√2cm
    60 cmssd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√21/√2= (1 ×√2) / (√2×√2) =√2/2side = 60 ×√2/2 = 30√2cm
    The side of the mirror is 60 × √22 = 30√2 cm.

  5. 5.(b) One side of the mirror is exactly 30√2 cm long, which is about 42.4 cm. Check: (30√2)2 = 900 × 2 = 1800, and two such squares add up to 3600 = 602.

    60 cm30√2cm30√2cmd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√21/√2= (1 ×√2) / (√2×√2) =√2/2side = 60 ×√2/2 = 30√2cmcheck: 2 × (30√2)2= 2 × 1800 = 3600 = 602
    60 cm30√2cm30√2cmd2= s2+ s2= 2s2, so d = s√2side / diagonal = s / (s√2) = 1/√21/√2= (1 ×√2) / (√2×√2) =√2/2side = 60 ×√2/2 = 30√2cmcheck: 2 × (30√2)2= 2 × 1800 = 3600 = 602
    (b) One side is exactly 30√2 cm long, which is about 42.4 cm.

Answer: (a) √22; (b) 30√2 cm

Common mistakes

  • Multiplying only the denominator by √2 and writing 12. That changes the value of the fraction. The numerator and the denominator must both be multiplied by √2, which gives √22.
  • Taking the side as half of the diagonal, 30 cm. A square with a side of 30 cm has a diagonal of 30√2 cm, which is about 42.4 cm and not 60 cm. The side is √22 of the diagonal, not 12 of it.

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