# Square Root of 21 — Value, How to Find It, and Examples TL;DR The square root of 21 is about $4.583$, and because $21 = 3 \times 7$ has no repeated prime factor, $\sqrt{21}$ is irrational and already in simplest radical form. This article gives the exact and decimal value, two by-hand methods, where $\sqrt{21}$ shows up, and the mistakes students make most. ## 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 | $n$ | $\sqrt{n}$ (exact) | $\sqrt{n}$ (4 d.p.) | | --- | --- | --- | | $9$ | $3$ | $3.0000$ | | $12$ | $2\sqrt{3}$ | $3.4641$ | | $15$ | $\sqrt{15}$ | $3.8730$ | | $16$ | $4$ | $4.0000$ | | $18$ | $3\sqrt{2}$ | $4.2426$ | | $20$ | $2\sqrt{5}$ | $4.4721$ | | **$21$** | $\boldsymbol{\sqrt{21}}$ | $\boldsymbol{4.5826}$ | | $22$ | $\sqrt{22}$ | $4.6904$ | | $24$ | $2\sqrt{6}$ | $4.8990$ | | $25$ | $5$ | $5.0000$ | | $27$ | $3\sqrt{3}$ | $5.1962$ | | $30$ | $\sqrt{30}$ | $5.4772$ | $\sqrt{21}$ 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. ## Frequently Asked Questions What is the value of the square root of 21? $\sqrt{21} \approx 4.583$, or $4.5825757\ldots$ to more places. Is the square root of 21 an irrational number? Yes. $21 = 3 \times 7$ is not a perfect square, so $\sqrt{21}$ cannot be written as a fraction and its decimal never repeats. What is the radical form of the square root of 21? $\sqrt{21}$ — it is already in simplest radical form, since there is no repeated prime factor to take out. Is 21 a perfect square? No. The nearest perfect squares are $16$ ($4^2$) and $25$ ($5^2$). What is the square root of 21 to two decimal places? $\sqrt{21} \approx 4.58$.