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# Square Root of 47 — Value, Simplification, and Steps

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

TL;DR

The square root of 47 is approximately 6.856, and it is irrational because 47 is a prime number with no square factors. This article shows why 47\sqrt{47}47​ is already in its simplest radical form, how to compute it by long division and estimation, and the errors to avoid.

BT

[Bhanzu Team](/content/authors/bhanzu-team/index.html) Last updated on July 20, 20265 min read

## What Is the Square Root of 47?

The square root of 47 is 47≈6.856\sqrt{47} \approx 6.85647​≈6.856. It is an **irrational number**, so the decimal runs forever without repeating and never settles into an exact fraction.

> **Quick Answer:**  
> **Result:** 47≈6.856\sqrt{47} \approx 6.85647​≈6.856  
> **Notation:** 47\sqrt{47}47​ (exact), 471/247^{1/2}471/2, or ≈6.856\approx 6.856≈6.856 (decimal)  
> **Method shown:** Long division and estimation  
> **Rational or irrational:** Irrational (47 is prime, not a perfect square)  
> **Exact form:** 47\sqrt{47}47​ (already simplest; no square factor to pull out)

Because 47 is not a perfect square, no whole number squares to it: 62=366^2 = 3662=36 and 72=497^2 = 4972=49, so 47\sqrt{47}47​ sits between 6 and 7, closer to 7.

## Quick Reference Table

The table places 47\sqrt{47}47​ among its neighbours and shows which roots are exact. Only perfect squares (49, 64) give whole-number roots.

| Number | Square root | Exact or approximate |
| --- | --- | --- |
| 45\sqrt{45}45​ | 35≈6.7083\sqrt{5} \approx 6.70835​≈6.708 | Approximate (irrational) |
| 46\sqrt{46}46​ | ≈6.782\approx 6.782≈6.782 | Approximate (irrational) |
| 47\sqrt{47}47​ | ≈6.856\approx 6.856≈6.856 | Approximate (irrational, prime) |
| 48\sqrt{48}48​ | 43≈6.9284\sqrt{3} \approx 6.92843​≈6.928 | Approximate (irrational) |
| 49\sqrt{49}49​ | 777 | Exact (perfect square) |
| 50\sqrt{50}50​ | 52≈7.0715\sqrt{2} \approx 7.07152​≈7.071 | Approximate (irrational) |
| 64\sqrt{64}64​ | 888 | Exact (perfect square) |

## Where the Square Root of 47 Shows Up

47\sqrt{47}47​ appears as the length of the diagonal of a rectangle whose sides multiply through Pythagoras to 47, for instance legs whose squares sum to 47. It also surfaces in physics and geometry any time a squared distance or a quadratic solution equals 47, where the answer must stay in exact radical form to avoid rounding error, then convert to ≈6.856\approx 6.856≈6.856 only at the end.

## What Does "Square Root" Mean Here?

A **square root** of a number is a value that multiplies by itself to give that number. Squaring and taking the [square root](/content/math/terms/square-root/index.html) are inverse operations.

A **prime number** has exactly two factors, 1 and itself, so 47 factors only as 1×471 \times 471×47. Because there is no repeated prime factor, nothing can be pulled out from under the radical, which is why 47\sqrt{47}47​ is already in simplest form.

### **Is the square root of 47 rational or irrational?**

It is **irrational**. A rational number is a ratio of two integers, but 47\sqrt{47}47​ cannot be written that way; its decimal is non-terminating and non-repeating. This is the same reason [√640](/content/math/algebra/square-root-of-640/index.html) is irrational, while a perfect square like [√4900](/content/math/algebra/square-root-of-4900/index.html) is rational.

## How to Compute the Square Root of 47

### **Method 1: Estimation between perfect squares**

Find the two nearest perfect squares.

62=366^2 = 3662=36 and 72=497^2 = 4972=49, so 6<47<76 < \sqrt{47} < 76<47​<7.

Since 47 is much closer to 49 than to 36, the root is close to 7.

Test 6.85: 6.852=46.92256.85^2 = 46.92256.852=46.9225, slightly low.

Test 6.86: 6.862=47.05966.86^2 = 47.05966.862=47.0596, slightly high.

So 47≈6.856\sqrt{47} \approx 6.85647​≈6.856.

**Final answer:** 47≈6.856\sqrt{47} \approx 6.85647​≈6.856

### **Method 2: Long division**

Write 47 as 47‾.00‾00‾\overline{47}.\overline{00}
\overline{00}47.0000 and pair digits around the decimal point.

The largest square under 47 is 36, and 36=6\sqrt{36} = 636​=6, so the first digit is 6; remainder 47−36=1147 - 36 = 1147−36=11.

Bring down 000000 to make 1100. Double the 6 to get 12, and find ddd with 12d×d≤110012d \times d \le 110012d×d≤1100: d=8d = 8d=8 gives 128×8=1024128 \times 8 = 1024128×8=1024.

The quotient is 6.8; remainder 1100−1024=761100 - 1024 = 761100−1024=76, bring down 000000 to make 7600.

Double 68 to get 136; 136d×d≤7600136d \times d \le 7600136d×d≤7600 needs d=5d = 5d=5, since 1365×5=68251365 \times 5 = 68251365×5=6825.

The quotient reads 6.85…6.85\ldots6.85…, refining to ≈6.856\approx 6.856≈6.856.

**Final answer:** 47≈6.856\sqrt{47} \approx 6.85647​≈6.856

## Common Mistakes With the Square Root of 47

### **Mistake 1: Trying to simplify the radical**

**Where it slips in:** A student expects every root to reduce, so they hunt for a factor to pull out of 47\sqrt{47}47​.

**Don't do this:** Write 47=4×something\sqrt{47} = \sqrt{4} \times \sqrt{something}47​=4​×something​. There is no perfect-square factor, since 47 is prime.

**The correct way:** Check for square factors first. With none, 47\sqrt{47}47​ is already the simplest exact form; only its decimal, ≈6.856\approx 6.856≈6.856, is an approximation.

### **Mistake 2: Rounding too early**

**Where it slips in:** In a longer calculation, a student replaces 47\sqrt{47}47​ with 6.9 in step one and carries the rounded value onward.

**Don't do this:** Substitute 6.9 and treat later results as exact. The error compounds.

**The correct way:** Keep 47\sqrt{47}47​ in radical form through the algebra and convert to ≈6.856\approx 6.856≈6.856 only in the final step.

### **Mistake 3: Confusing 47\sqrt{47}47​ with 47247^2472**

**Where it slips in:** Reading fast, students square 47 instead of rooting it.

**Don't do this:** Answer 2209 for 47\sqrt{47}47​. That is 47247^2472, the opposite operation.

**The correct way:** The square root asks what number times itself gives 47, which is about 6.856, not 2209.

## Conclusion

- The **square root of 47 is approximately 6.856** and is irrational.

- 47 is prime, so 47\sqrt{47}47​ has no square factor and is already in simplest radical form.

- It sits between 6 and 7, closer to 7, because 47 is near the perfect square 49.

- Long division and estimation both reach ≈6.856\approx 6.856≈6.856.

- Keep 47\sqrt{47}47​ exact through a calculation and round only at the end.