Square Root of 5000 - Value, Simplest Form, and Examples

What Is A Square Root?

A square root of a number $n$ is a value $r$ such that $r^2 = n$. The square root of 5000 is the positive number that, multiplied by itself, gives 5000.

No whole number works, because $70^2 = 4900$ is too small and $71^2 = 5041$ is too big. So $\sqrt{5000}$ lands between 70 and 71, and because 5000 is very close to 5041, the value sits just above 70.7.

Where Does √5000 Appear?

$\sqrt{5000}$ is the diagonal of a square whose side is 50 units, because a square of side $s$ has diagonal $s\sqrt{2}$, and $50\sqrt{2} = \sqrt{5000}$. It also turns up in area work: a square of area 5000 square units has each side equal to $\sqrt{5000} = 50\sqrt{2}$ units.

Quick Reference Table

Is The Square Root Of 5000 Rational Or Irrational?

$\sqrt{5000}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of integers, and its decimal never terminates or repeats.

A whole number has a rational square root only when it is a perfect square. Since 5000 falls strictly between $70^2$ and $71^2$, it is not a perfect square, so its root is an irrational number.

The exact value is best kept as $50\sqrt{2}$, because the irrational part is entirely captured by $\sqrt{2}$, one of the most studied irrational numbers in mathematics. See Wikipedia on the square root of 2 for its long history.

How Do You Find √5000? (Prime Factorization And Long Division)

Prime factorization (for the exact simplest form).

$$5000 = 2^3 \times 5^4$$

$$5000 = (2^2 \times 5^4) \times 2$$

$$\sqrt{5000} = \sqrt{2^2 \times 5^4} \times \sqrt{2}$$

$$\sqrt{5000} = (2 \times 5^2)\sqrt{2}$$

$$\sqrt{5000} = 50\sqrt{2}$$

Because $\sqrt{2}$ has no square factor, $50\sqrt{2}$ is fully simplified. This is the same extraction taught for simplifying radical expressions.

Long division (for the decimal value).

Step 1: Pair the digits around the decimal point: $\overline{50}\ \overline{00}.\overline{00}\ \overline{00}$.

Step 2: The largest square $\leq 50$ is $49$ ($7^2 = 49$). First quotient digit is $7$; remainder $1$.

Step 3: Bring down $00$ to get $100$. Double $7 \to 14$. Find $d$ with $(140 + d),d \leq 100$; $d = 0$ works. Quotient $70$, remainder $100$.

Step 4: Bring down $00$ to get $10000$. Double $70 \to 140$. Find $d$ with $(1400 + d),d \leq 10000$; $d = 7$ gives $1407 \times 7 = 9849$. Quotient $70.7$, remainder $151$.

Step 5: Bring down $00$ to get $15100$. Double $707 \to 1414$. Find $d$ with $(14140 + d),d \leq 15100$; $d = 1$ gives $14141 \times 1 = 14141$. Quotient $70.71$, remainder $959$.

Step 6: Continue two more places to reach $\sqrt{5000} \approx 70.7107$, which never terminates.

Examples Of √5000

Example 1

Simplify $\sqrt{5000}$ to simplest radical form.

$$\sqrt{5000} = \sqrt{2500 \times 2}$$

$$\sqrt{5000} = \sqrt{2500} \times \sqrt{2}$$

$$\sqrt{5000} = 50\sqrt{2}$$

Final answer: $50\sqrt{2}$.

Example 2

Simplify $\sqrt{5000}$ by first spotting the factor 100. First instinct, then the check.

A natural first move is $\sqrt{5000} = \sqrt{100 \times 50} = 10\sqrt{50}$, then stop. Look closer at $\sqrt{50}$: it still hides a square, since $50 = 25 \times 2$. Stopping at $10\sqrt{50}$ leaves the radical only half-simplified.

Finish the job.

$$10\sqrt{50} = 10 \times \sqrt{25 \times 2}$$

$$10\sqrt{50} = 10 \times 5\sqrt{2}$$

$$10\sqrt{50} = 50\sqrt{2}$$

The lesson is to keep extracting squares until nothing square is left under the radical.

Example 3

Confirm that squaring the simplified form returns 5000.

$$(50\sqrt{2})^2 = 50^2 \times (\sqrt{2})^2$$

$$(50\sqrt{2})^2 = 2500 \times 2 = 5000$$

Example 4

Evaluate $\dfrac{\sqrt{5000}}{\sqrt{2}}$.

$$\frac{\sqrt{5000}}{\sqrt{2}} = \sqrt{\frac{5000}{2}}$$

$$\frac{\sqrt{5000}}{\sqrt{2}} = \sqrt{2500} = 50$$

Example 5

A square has side 50 units. How long is its diagonal?

$$d = 50\sqrt{2}$$

$$d = \sqrt{5000} \approx 70.71 \text{ units}$$

Common Mistakes

Mistake 1: Stopping before the radical is fully simplified

Where it slips in: Pulling out an easy square like 100 and leaving $10\sqrt{50}$.

Don't do this: Reporting $\sqrt{5000} = 10\sqrt{50}$ as the final answer.

The correct way: $\sqrt{50}$ still contains the square 25, so keep going to $50\sqrt{2}$. The rusher who grabs the first square and stops leaves the answer incomplete every time the leftover radical is itself reducible.

Mistake 2: Moving a factor out without square-rooting it

Where it slips in: Extracting 2500 from under the radical.

Don't do this: Writing $\sqrt{5000} = \sqrt{2500 \times 2} = 2500\sqrt{2}$.

The correct way: The 2500 leaves as its own square root: $\sqrt{2500} = 50$, so $\sqrt{5000} = 50\sqrt{2}$, not $2500\sqrt{2}$.

Mistake 3: Rounding too early

Where it slips in: Long problems where $\sqrt{5000}$ appears mid-calculation.

Don't do this: Replace $\sqrt{5000}$ with 70.71 at the start and carry that through.

The correct way: Keep the exact $50\sqrt{2}$ until the final step, then round once, so error does not accumulate.

Conclusion

The square root of 5000 is $50\sqrt{2}$, roughly $70.7107$, and it is irrational because 5000 is not a perfect square. Prime factorization gives the exact form; long division gives the decimal; the trick is to keep extracting squares until only $\sqrt{2}$ remains. To go deeper into radicals with a teacher, explore Bhanzu's algebra tutor or a high school math tutor, or browse math classes online. Want to try one live? Book a free demo class.

Read More

• Square Root 1 to 30 — every root from 1 to 30 in one reference table.

• Squares and Square Roots — how squaring and rooting reverse each other.

• Square Root of 50 — the $5\sqrt{2}$ root that $\sqrt{5000}$ is built on.

• Square Root of 2000 — another large root that simplifies, to $20\sqrt{5}$.

• Square Root Tricks — fast estimation for large non-perfect squares.