Square roots of 1 to 200

The rounded decimal and the exact simplified surd, side by side: √50 is 7.071068 and also 5√2.

The fourteen perfect squares

Only fourteen of them come out whole below 200. The other 186 are irrational — their decimals never end.

√1 = 1√4 = 2√9 = 3√16 = 4√25 = 5√36 = 6√49 = 7√64 = 8√81 = 9√100 = 10√121 = 11√144 = 12√169 = 13√196 = 14√225 = 15

Roots that simplify

These have a square factor to pull out. On an exam this form, not 7.07, is the right answer.

√8 = 2√2√12 = 2√3√18 = 3√2√20 = 2√5√24 = 2√6√27 = 3√3√28 = 2√7√32 = 4√2√40 = 2√10√44 = 2√11√45 = 3√5√48 = 4√3√50 = 5√2√52 = 2√13√54 = 3√6√56 = 2√14√60 = 2√15√63 = 3√7√68 = 2√17√72 = 6√2√75 = 5√3√76 = 2√19√80 = 4√5√84 = 2√21√88 = 2√22√90 = 3√10√92 = 2√23√96 = 4√6√98 = 7√2√99 = 3√11√104 = 2√26√108 = 6√3√112 = 4√7√116 = 2√29√117 = 3√13√120 = 2√30√124 = 2√31√125 = 5√5√126 = 3√14√128 = 8√2√132 = 2√33√135 = 3√15√136 = 2√34√140 = 2√35√147 = 7√3√148 = 2√37√150 = 5√6√152 = 2√38√153 = 3√17√156 = 2√39√160 = 4√10√162 = 9√2√164 = 2√41√168 = 2√42√171 = 3√19√172 = 2√43√175 = 5√7√176 = 4√11√180 = 6√5√184 = 2√46√188 = 2√47√189 = 3√21√192 = 8√3√198 = 3√22√200 = 10√2√204 = 2√51√207 = 3√23√208 = 4√13√212 = 2√53√216 = 6√6√220 = 2√55√224 = 4√14√228 = 2√57

How the surd is simplified

Pull out the largest square factor. Since 50 = 25 × 2, the root of 25 comes out as 5 and the 2 stays behind: 5√2.

From 1 to 200

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230

How to read this

The decimal is rounded; the surd form is the exact value.
Pulling out a square factor simplifies the surd — √50 = 5√2.
If the number is not a perfect square, the decimal never ends (irrational).
Knowing which two integers it sits between is enough to estimate in your head.

Frequently asked questions

QWhy is √50 equal to 5√2?

Split 50 into 25 × 2. Since 25 is a square, its root 5 comes out; the 2 has no square factor left, so it stays under the sign.

QCan I just use the decimal?

It depends. For measuring, 7.07 is easier; in a maths answer the unrounded 5√2 is what counts as correct.

QWhat does irrational mean?

It cannot be written as a fraction, and its decimal neither ends nor repeats. Any square root of a non-square is like this.

QHow do I estimate one in my head?

Find the nearest perfect square. For √50, 49 is 7², so the answer is just above 7.

QWhy stop at 200?

Because the numbers people look up are mostly two-digit, or the ones from the table like 100, 144 and 169.