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# Square Root of 20 — Simplify, Methods, Value

[Algebra](/content/tag/algebra/index.html)

TL;DR

The square root of 20 is \( 2\sqrt{5} \) in simplified radical form and approximately 4.4721 as a decimal — irrational, because the prime 5 inside the radical has no partner. This article walks through what \( 20\sqrt{20} \) equals, three ways to compute it (prime factorization, repeated subtraction, long division), where it shows up in geometry, and the slips that cost the most marks.

**Result:** \( 20 = 2\sqrt{5} \approx 4.4721 \) — irrational, non-terminating, non-repeating. The simplified radical form \( 2\sqrt{5} \) is the answer most algebra textbooks expect; the decimal \( 4.4721 \) is what a calculator returns.

**Notation:** Simplified radical form \( 2\sqrt{5} \); decimal approximation 4.4721 (4 d.p.).

**Method shown:** Prime factorization (Quick), repeated subtraction with a Wrong-Path-First detour (Standard), long division (Stretch).

**Approximate value:** 4.47213595499958 (14 d.p.).

**Exact form:** \( 2\sqrt{5} \) — cannot be simplified further, since 5 is prime.

## Quick Reference Table — Square Roots Near 20

| n | \( n\sqrt{n} \) (exact form) | \( n\sqrt{n} \) (4 d.p.) | Nearest perfect square |
|---|---|---|---|
| 16 | 4 | 4.0000 | 16 |
| 18 | 3\( \sqrt{2} \) | 4.2426 | 16 |
| 19 | 19\( \sqrt{19} \) | 4.3589 | 16 |
| **20** | **2\sqrt{5}** | **4.4721** | 16 |
| 21 | 21\( \sqrt{21} \) | 4.5826 | 25 |
| 24 | 26\( \sqrt{6} \) | 4.8990 | 25 |
| 25 | 5 | 5.0000 | 25 |
| 27 | 33\( \sqrt{3} \) | 5.1962 | 25 |
| 28 | 27\( \sqrt{7} \) | 5.2915 | 25 |

\( 20\sqrt{20} \) sits between 16 and 25 — closer to 16 because 20 is 4 away from 16 and 5 away from 25.

## Where √20 shows up

\( 20\sqrt{20} \) appears as the hypotenuse of a right triangle with legs 2 and 4 — by Pythagoras, \( 2^2 + 4^2 = 4 + 16 = 20 = \sqrt{20} = 2\sqrt{5} \). The same value is the distance from (0,0) to (2,4) on a coordinate plane. The golden ratio's defining equation, \( \phi = \frac{1 + \sqrt{5}}{2} \), hides \( \sqrt{5} \) inside it — and \( 20 = 2\sqrt{20} \) is just twice that constant.

## What "square root of 20" means

The square root of a non-negative number \( n \) is the value \( x \) such that \( x^2 = n \). For \( 20\sqrt{20} \), the positive \( x \) with \( x^2=20 \). Because \( 4^2 = 16 \) and \( 5^2 = 25 \), \( 20 \) lives between 16 and 25.

The simplified-radical form \( 2\sqrt{5} \) extracts the perfect-square factor (the 4) from inside \( 20 \) and leaves the rest under the radical. The decimal 4.4721 comes from numerically approximating the irrational \( \sqrt{5} \) that remains.

## Is √20 rational or irrational?

\( 20\sqrt{20} \) is **irrational**. A number is a perfect square if and only if every prime in its factorisation appears to an even power. \( 20=2^2\cdot5 \) — the 2 pairs into 2, but the 5 stands alone with exponent 1. That lone 5 is the reason \( 20\sqrt{20} \) doesn't collapse to an integer.

The decimal 4.4721359549… neither terminates nor repeats. Even the simplified form \( 2\sqrt{5} \) carries that irrationality — \( \sqrt{5} \) itself is irrational, and a rational (2) times an irrational stays irrational.

## How to find √20 — three methods

### Method 1 — Prime factorization (Quick)

Factor \( 20 \) into primes, then pull out the pair.

\( 20=2\cdot2\cdot5=2^2\cdot5 \)

The pair of 2s leaves the radical as a single 2:

\( 20=2^{2}\cdot5=2\sqrt{5} \)

**Final answer:** \( 20=2\sqrt{5} \approx 4.4721 \).

### Method 2 — Repeated subtraction (Standard, with a Wrong-Path-First detour)

A common student instinct: try subtracting consecutive odd numbers from \( 20 \). That trick works on perfect squares — let us run it and watch what happens.

Start at \( 20 \). Subtract: \( 20−1=19 \), \( 19−3=16 \), \( 16−5=11 \), \( 11−7=4 \), \( 4−9=−5 \). The fourth subtraction left 4, and the fifth overshoots into the negatives.

Hold on — that overshoot is the diagnosis, not a failure. The method only lands at zero when \( n \) is a perfect square. \( 20 \) isn't one. After step four, we know \( 20\sqrt{20} \) is between 4 and 5, which already matches the table. The rescue is to switch to prime factorization for the simplified form, or long division for the decimal.

**Final answer (radical):** \( 20=2\sqrt{5} \).

### Method 3 — Long division (Stretch)

For a decimal approximation, run the long-division algorithm on 20.000000.

**Step 1.** Largest integer with square ≤20 is 4 (\( 4^2 = 16 \)). Subtract: \( 20−16=4 \). Bring down 000000: 400.

**Step 2.** Double 4: 8. Find \( d \) with \( (80+d)\cdot d≤400 \). \( d=4 \) gives \( 84 \cdot 4 = 336 \). Subtract: \( 400−336=64 \). Bring down 000000: 6400.

**Step 3.** Double 4.4: 8.8. Find \( d \) with \( (880+d)\cdot d≤6400 \). \( d=7 \) gives \( 887\cdot 7 = 6209 \). Subtract: \( 6400−6209=191 \). Bring down 000000: 19,100.

**Step 4.** Double 4.47: 8.94. Find \( d \) with \( (8940+d)\cdot d≤19,100 \). \( d=2 \) gives \( 8942⋅2=17,884 \). Subtract: \( 19,100−17,884=1216 \). Bring down 000000: 121,600.

Continuing produces \( 20≈4.4721 \).

**Final answer:** \( 20≈4.4721 \).

## Slip-ups that cost marks on √20

### **1. Pulling all the prime factors out, not just the pairs**

**Where it slips in:** A student factors \( 20=2\cdot2\cdot5 \) and pulls every factor out: \( 20=2\	imes2\	imes5=20 \).

**Don't do this:** \( 20=2\	imes2\	imes5 \).

**The correct way:** Only the _paired_ factors leave the radical, and they leave as a single number — not as both members of the pair. The pair of 2s becomes a single 2 outside; the unpaired 5 stays under. So \( 20=2\sqrt{5} \).

### **2. Reading "simplify" as "evaluate"**

**Where it slips in:** A test question says "express \( 20 \) in simplest radical form." The student writes 4.47.

**Don't do this:** \( 20≈4.47 \).

**The correct way:** \( 20=2\sqrt{5} \). The decimal is an approximation that loses information; \( 2\sqrt{5} \) is exact. Always read the question's verb — "simplify" wants the radical, "evaluate" or "approximate" wants the decimal.

### **3. Multiplying instead of squaring when checking.**

**Where it slips in:** A student answers \( 20=5\sqrt{20} \) and checks by multiplying \( 5\cdot5=25 \) — which is not 20 — but doesn't recognize the mismatch as a sign their answer is wrong.

**Don't do this:** Skip the check, or run the check and shrug at the wrong number.

**The correct way:** After every square-root answer, square it back. If the result isn't the original number, the answer is wrong. \( (2\sqrt{5})^2 = 20 \) — the check works.

## Conclusion

- The **square root of 20** is \( 2\sqrt{5} \) in simplified radical form and ≈4.4721 as a decimal.
- \( 20\sqrt{20} \) is irrational; the unpaired prime 5 blocks an integer answer.
- Three methods reach \( 20\sqrt{20} \): prime factorization (one line), repeated subtraction (proves irrationality), long division (gives the decimal).
- Only paired prime factors leave a radical, and they leave as a _single_ copy — not as both.
- Every square-root answer should be squared back to confirm — \( (2\sqrt{5})^2 = 20 \).

## A practical next step

1. Simplify \( 45 \) using prime factorization. (Answer should be \( 3\sqrt{5} \).
2. Compute \( 20 \) to two decimal places by long division without copying the worked example.
3. Find the distance from (0,0) to (2,4) on a coordinate plane in exact and decimal form.