Square Root of 47 — Value, Simplification, and Steps
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Square Root of 47 — Value, Simplification, and Steps
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.
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Bhanzu Team 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 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 is irrational, while a perfect square like √4900 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.