What Does Square Root 1 To 30 Mean?
The square root of a number $n$ is the value $r$ with $r \times r = n$. So $\sqrt{9} = 3$ because $3^2 = 9$, and $\sqrt{16} = 4$ because $4^2 = 16$.
Across 1 to 30, only five numbers are perfect squares - whole numbers that come from squaring another whole number. The other 25 are not, so their roots are irrational numbers: decimals that run forever without repeating.
Every root here is a positive value. The radical symbol $\sqrt{\ }$ always names the principal (non-negative) root, even though $x^2 = n$ has both a positive and a negative solution.
What Are The Values Of Square Roots From 1 To 30?
The chart below gives each root three ways: exact notation, simplified radical form (a smaller number pulled out where a perfect-square factor exists), and the decimal to four places.
$n$ | Exact | Simplified | Decimal | Type |
|---|---|---|---|---|
1 | $\sqrt{1}$ | 1 | 1.0000 | Rational |
2 | $\sqrt{2}$ | $\sqrt{2}$ | 1.4142 | Irrational |
3 | $\sqrt{3}$ | $\sqrt{3}$ | 1.7321 | Irrational |
4 | $\sqrt{4}$ | 2 | 2.0000 | Rational |
5 | $\sqrt{5}$ | $\sqrt{5}$ | 2.2361 | Irrational |
6 | $\sqrt{6}$ | $\sqrt{6}$ | 2.4495 | Irrational |
7 | $\sqrt{7}$ | $\sqrt{7}$ | 2.6458 | Irrational |
8 | $\sqrt{8}$ | $2\sqrt{2}$ | 2.8284 | Irrational |
9 | $\sqrt{9}$ | 3 | 3.0000 | Rational |
10 | $\sqrt{10}$ | $\sqrt{10}$ | 3.1623 | Irrational |
11 | $\sqrt{11}$ | $\sqrt{11}$ | 3.3166 | Irrational |
12 | $\sqrt{12}$ | $2\sqrt{3}$ | 3.4641 | Irrational |
13 | $\sqrt{13}$ | $\sqrt{13}$ | 3.6056 | Irrational |
14 | $\sqrt{14}$ | $\sqrt{14}$ | 3.7417 | Irrational |
15 | $\sqrt{15}$ | $\sqrt{15}$ | 3.8730 | Irrational |
16 | $\sqrt{16}$ | 4 | 4.0000 | Rational |
17 | $\sqrt{17}$ | $\sqrt{17}$ | 4.1231 | Irrational |
18 | $\sqrt{18}$ | $3\sqrt{2}$ | 4.2426 | Irrational |
19 | $\sqrt{19}$ | $\sqrt{19}$ | 4.3589 | Irrational |
20 | $\sqrt{20}$ | $2\sqrt{5}$ | 4.4721 | Irrational |
21 | $\sqrt{21}$ | $\sqrt{21}$ | 4.5826 | Irrational |
22 | $\sqrt{22}$ | $\sqrt{22}$ | 4.6904 | Irrational |
23 | $\sqrt{23}$ | $\sqrt{23}$ | 4.7958 | Irrational |
24 | $\sqrt{24}$ | $2\sqrt{6}$ | 4.8990 | Irrational |
25 | $\sqrt{25}$ | 5 | 5.0000 | Rational |
26 | $\sqrt{26}$ | $\sqrt{26}$ | 5.0990 | Irrational |
27 | $\sqrt{27}$ | $3\sqrt{3}$ | 5.1962 | Irrational |
28 | $\sqrt{28}$ | $2\sqrt{7}$ | 5.2915 | Irrational |
29 | $\sqrt{29}$ | $\sqrt{29}$ | 5.3852 | Irrational |
30 | $\sqrt{30}$ | $\sqrt{30}$ | 5.4772 | Irrational |
How Do You Read And Use The Square Root Table?
Read each row left to right: the number $n$, its root in exact notation, the tidiest form to write it, and the decimal you would use in a calculation. For $\sqrt{20}$, the row says $2\sqrt{5} \approx 4.4721$. Carry the radical form in algebra, and swap in the decimal only when you need a number.
How do you memorize the square roots 1 to 30?
You do not memorize all 30. Lock in the five perfect-square roots (1, 2, 3, 4, 5), then estimate everything else from its neighbours. $\sqrt{18}$ sits between $\sqrt{16} = 4$ and $\sqrt{25} = 5$, and since 18 is close to 16, the answer is a little above 4 (it is 4.24). That single habit rebuilds any value on the chart without rote recall.
Which Square Roots From 1 To 30 Are Rational?
Exactly five: $\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}$, the roots of the perfect squares $1, 4, 9, 16, 25$. Their values are the whole numbers 1, 2, 3, 4, 5.
The pattern is worth seeing directly. Perfect squares come from $1^2, 2^2, 3^2, 4^2, 5^2$, and the gap between consecutive squares grows by 2 each time:
$$1,\ 4,\ 9,\ 16,\ 25$$ $$\text{gaps: } 3,\ 5,\ 7,\ 9$$
Every other number from 1 to 30 falls between two perfect squares, so its root is trapped between two whole numbers and turns out irrational. That is why 25 of the 30 roots never resolve to a clean fraction.
How Do You Compute A Square Root From 1 To 30?
Three methods cover everything on the chart. What are the different methods to find the square root of a number? Estimation, prime factorization, and long division.
Estimation between perfect squares. For $\sqrt{14}$:
$$9 < 14 < 16$$ $$3 < \sqrt{14} < 4$$
Because 14 is close to 16, the root is near the top of that range, about 3.74.
Prime factorization (for simplifying, not decimals). For $\sqrt{18}$:
$$18 = 2 \times 3^2$$ $$\sqrt{18} = 3\sqrt{2}$$
Long division (for decimals of any root). Pair digits from the decimal point, find the largest square below the leading group, then repeat the double-and-test loop. The full walk-through for a single number lives in square root tricks and in the step-by-step long division guide, and the underlying idea is the classical definition of square roots.
Examples Of Square Root 1 To 30
Example 1
Read $\sqrt{25}$ from the chart and confirm it.
$$5 \times 5 = 25$$ $$\sqrt{25} = 5$$
Final answer: 5, a rational value.
Example 2
Where estimation goes wrong. To place $\sqrt{20}$, a common guess is to average the endpoints of $4 < \sqrt{20} < 5$ and call it $\sqrt{20} \approx 4.5$. Does that check out? Squaring gives $4.5^2 = 20.25$, which is already past 20, so 4.5 is a touch too high. The midpoint only works when $n$ sits exactly halfway; here 20 is closer to 16 than to 25, so the root leans low, at 4.47.
Final answer: $\sqrt{20} \approx 4.47$, not 4.5.
Example 3
Simplify $\sqrt{24}$.
$$24 = 2^3 \times 3 = 2^2 \times 6$$ $$\sqrt{24} = 2\sqrt{6}$$
Final answer: $2\sqrt{6} \approx 4.8990$.
Example 4
Order $\sqrt{12}$, $\sqrt{7}$, and $\sqrt{21}$ from smallest to largest.
$$\sqrt{7} \approx 2.65,\quad \sqrt{12} \approx 3.46,\quad \sqrt{21} \approx 4.58$$
Final answer: $\sqrt{7} < \sqrt{12} < \sqrt{21}$.
Example 5
Which is closer to a whole number, $\sqrt{26}$ or $\sqrt{30}$?
$$\sqrt{26} \approx 5.099 \ (\text{0.099 above } 5)$$ $$\sqrt{30} \approx 5.477 \ (\text{0.477 from } 5\text{, } 0.523 \text{ from } 6)$$
Final answer: $\sqrt{26}$, which sits just above the perfect square 25.
Common Mistakes
Mistake 1: Assuming every root simplifies
Where it slips in: Trying to pull a factor out of a prime radicand like $\sqrt{13}$ or $\sqrt{17}$.
Don't do this: Writing $\sqrt{13}$ as some smaller multiple of a root.
The correct way: A radical simplifies only when the number has a perfect-square factor. $8, 12, 18, 20, 24, 27, 28$ do; the primes and square-free numbers do not. The habit that saves time is checking the factor list first, before reaching for a rewrite.
Mistake 2: Confusing the square with the square root
Where it slips in: Reading the chart quickly.
Don't do this: Writing $\sqrt{16} = 256$ by squaring instead of rooting.
The correct way: $\sqrt{16} = 4$ because $4^2 = 16$. Squaring and square-rooting are opposite operations; the root is the smaller number that builds $n$.
Mistake 3: Rounding the decimal too soon
Where it slips in: Using a two-place decimal in a multi-step problem.
Don't do this: Replace $\sqrt{30}$ with 5.5 and carry it forward.
The correct way: Keep at least four places (5.4772), or better, keep the exact radical until the final step.
Conclusion
The square root 1 to 30 chart holds five rational roots (1, 2, 3, 4, 5) and 25 irrational ones, with seven of those simplifying to a smaller radical. Memorize the perfect squares, estimate the rest from their neighbours, and reach for long division only when you need an exact decimal.
To build fluency with roots and radicals alongside a teacher, explore Bhanzu's algebra tutor, our high school math tutor sessions, or math classes online. Ready to practise a set live? Book a free demo class.
Read More
Square root of 3 — a core irrational root worth knowing by heart.
Square root of 5 — the value behind the golden ratio.
Square root of 10 — a worked non-perfect root with long division.
Square root of 20 — how $2\sqrt{5}$ is simplified in full.
Square root of 25 — a perfect-square root explained.
Perfect squares — the five numbers that make the chart's rational roots.
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