Square Root of 21 — Value, How to Find It, and Examples

The Answer At A Glance

Result: $\sqrt{21} \approx 4.5825757$

Notation: Decimal approximation; exact form is $\sqrt{21}$.

Method shown: Long division by hand, cross-checked with the average (Babylonian) method.

Approximate value: $4.5826$ (4 d.p.)

Exact form: $\sqrt{21}$ — cannot be simplified, since $21 = 3 \times 7$ has no square factor.

Quick Reference Table — Square Roots From 9 to 30

$\sqrt{21}$ sits between $\sqrt{16} = 4$ and $\sqrt{25} = 5$, just past the midpoint because $21$ is closer to $25$ than to $16$.

Where √21 Appears

$\sqrt{21}$ is the hypotenuse of a right triangle with legs $\sqrt{5}$ and $4$, since $\sqrt{(\sqrt{5})^2 + 4^2} = \sqrt{5 + 16} = \sqrt{21}$. It also turns up as the geometric mean of $3$ and $7$ — the side of a square whose area equals a $3 \times 7$ rectangle, since $\sqrt{3 \times 7} = \sqrt{21}$. That geometric-mean role is why $\sqrt{21}$ appears in scaling and proportion problems where a single length has to "average" two others multiplicatively.

What "square root of 21" Means

The square root of a non-negative number $n$ is the value $x$ such that $x^2 = n$. For $\sqrt{21}$, it is the positive $x$ with $x^2 = 21$.

Because $4^2 = 16$ and $5^2 = 25$, the answer must land between $4$ and $5$ — and $4.583^2 = 21.0039\ldots$, which confirms the value.

Is The Square Root of 21 Rational or Irrational?

$\sqrt{21}$ is irrational. Its prime factorisation is $21 = 3 \times 7$ — two distinct primes, each appearing once, so no prime sits at an even power and $21$ is not a perfect square.

With no repeated factor to pull out, $\sqrt{21}$ also cannot be simplified into a smaller radical. The decimal $4.5825757\ldots$ never terminates and never repeats — it cannot be written as a fraction $p/q$.

How To Find √21 — Two Methods

Method 1 — Long division (digit by digit)

Write $21$ as $21.000000$ and pair the digits.

Step 1. The largest integer whose square is at most $21$ is $4$ ($4^2 = 16$). Subtract: $21 - 16 = 5$. Bring down $00$ to get $500$.

Step 2. Double the quotient $4$ to get $8$. Find $d$ with $(80 + d)\cdot d \leq 500$. Here $d = 5$ gives $85 \cdot 5 = 425$. Subtract: $500 - 425 = 75$. Bring down $00$ to get $7500$.

Step 3. Double $4.5$ to get $90$. Find $d$ with $(900 + d)\cdot d \leq 7500$. Here $d = 8$ gives $908 \cdot 8 = 7264$. Subtract: $7500 - 7264 = 236$.

Continuing produces $4.5825\ldots$

Final answer: $\sqrt{21} \approx 4.5826$.

Method 2 — Average (Babylonian) method

Start with a guess $x_0 = 4.6$, then average it with $21$ divided by the guess:

$$x_{k+1} = \frac{1}{2}\left(x_k + \frac{21}{x_k}\right)$$

• $x_1 = \frac{1}{2}\left(4.6 + \frac{21}{4.6}\right) = \frac{1}{2}(4.6 + 4.5652) = 4.5826$

One step already lands on $4.5826$, because the starting guess was close.

Final answer: $\sqrt{21} \approx 4.5826$.

What are the most common mistakes with √21?

Mistake 1: Trying to simplify a non-square radicand

Where it slips in: A student factorises $21 = 3 \times 7$ and then tries to take one factor out of the root.

Don't do this: $\sqrt{21} = \sqrt{3},\sqrt{7} = 3\sqrt{7}$ or $7\sqrt{3}$.

The correct way: A factor leaves the radical only when it appears as a pair. Here $3$ and $7$ each appear once, so nothing comes out — $\sqrt{21}$ is already simplest.

Mistake 2: Splitting the root over addition

Where it slips in: When $\sqrt{21}$ appears as $\sqrt{5 + 16}$ in a Pythagoras calculation.

Don't do this: $\sqrt{5 + 16} = \sqrt{5} + \sqrt{16} = 2.236 + 4 = 6.236$.

The correct way: $\sqrt{5 + 16} = \sqrt{21} \approx 4.583$. Square roots distribute over multiplication, never over addition.

Mistake 3: Reporting a negative value as "the" square root

Where it slips in: Recalling that $x^2 = 21$ has two solutions and writing the principal root as negative.

Don't do this: Claiming $\sqrt{21} = -4.583$.

The correct way: The radical symbol means the principal (non-negative) root, so $\sqrt{21} \approx +4.583$. The equation $x^2 = 21$ has both $+\sqrt{21}$ and $-\sqrt{21}$, but $\sqrt{21}$ alone is positive.

Examples of Square Root of 21

Example 1

Confirm that $\sqrt{21}$ does not simplify.

$21 = 3 \times 7$, two single primes, so $\sqrt{21}$ stays as is — about $4.583$.

Example 2 (Wrong path first)

Find the side of a square whose area equals a $3 \times 7$ rectangle.

Wrong attempt. A student averages the sides: $\frac{3 + 7}{2} = 5$.

Why it breaks. A $5 \times 5$ square has area $25$, but the rectangle's area is $3 \times 7 = 21$ — the squares don't match.

Correct. The matching side is the geometric mean: $\sqrt{3 \times 7} = \sqrt{21} \approx 4.583$, and $4.583^2 \approx 21$.

Example 3

Evaluate $\sqrt{21} \times \sqrt{21}$.

$\sqrt{21} \times \sqrt{21} = 21$. The square root and the square cancel exactly.

Example 4

Simplify $\sqrt{21} \times \sqrt{3}$.

$\sqrt{21} \times \sqrt{3} = \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \approx 7.937$. Multiplying combines the radicands, then a square factor ($9$) appears.

Example 5

A square plot has area $21$ square metres. Find its side length.

Side $= \sqrt{21} \approx 4.58$ m. The whole-number area still gives an irrational side, since $21$ is not a perfect square.

Conclusion

• The square root of 21 is approximately $4.583$ — irrational, non-terminating, non-repeating.

• $21 = 3 \times 7$ has no repeated prime, so $\sqrt{21}$ cannot be simplified.

• Long division and the average method both reach the value by hand.

• A factor leaves a radical only when it appears as a pair — single primes stay inside.

• $\sqrt{21}$ is the geometric mean of $3$ and $7$, the side of a square equal in area to a $3 \times 7$ rectangle.

A practical next step

1. Find $\sqrt{22}$ to three decimal places by long division and check by squaring.

2. Show that $\sqrt{20}$ simplifies to $2\sqrt{5}$ while $\sqrt{21}$ does not simplify.

3. Use the average method from a starting guess of $4.5$ to recover $\sqrt{21}$ to four decimals.

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