Square Root Of 52: Value, Simplified Form And Long Division

What Is A Square Root?

A square root of a number $n$ is a value $r$ with $r^2 = n$. Every positive number has two square roots, one positive and one negative, so $52$ has $2\sqrt{13}$ and $-2\sqrt{13}$.

The symbol $\sqrt{52}$ always means the positive one, called the principal square root. The number under the sign, here $52$, is the radicand.

A perfect square is a whole number that is the square of a whole number, like $49 = 7^2$ or $64 = 8^2$. Since $52$ falls strictly between them, $\sqrt{52}$ lies strictly between $7$ and $8$, and it is closer to $7$.

Where Does Root 52 Show Up, And Why Does It Matter?

Root 52 is the diagonal of any 4 by 6 rectangle, because the Pythagoras theorem gives $\sqrt{4^2 + 6^2} = \sqrt{52}$. A standard 4 by 6 inch photo print therefore measures $2\sqrt{13} \approx 7.21$ inches corner to corner.

The same number is the gap between the points $(1, 2)$ and $(5, 8)$ on a grid, since the distance between two points is $\sqrt{4^2 + 6^2}$. It is also the side of a square whose area is $52$ square units, and the radius of a circle whose area is $52\pi$.

Quick Reference Table

The table lists square roots of nearby numbers, so you can see where $\sqrt{52}$ sits and which neighbours simplify.

The last two rows share the factor $\sqrt{13}$ with $\sqrt{52}$, which is why they combine neatly in Example 4. The full worked page for the neighbour square root of 50 uses the same pull-out-a-square idea.

What Is The Square Root Of 52 In Simplest Radical Form?

The square root of 52 in simplest radical form is $2\sqrt{13}$. To simplify, pull out the largest perfect-square factor of $52$, which is $4$:

$$\sqrt{52} = \sqrt{4 \times 13}$$

$$= \sqrt{4} \times \sqrt{13}$$

$$= 2\sqrt{13}$$

The leftover $13$ is prime, so it has no square factor to remove and $2\sqrt{13}$ cannot be simplified further. The general rule, $\sqrt{ab} = \sqrt{a}\sqrt{b}$ for non-negative $a$ and $b$, is covered in simplifying radical expressions.

Is The Square Root Of 52 Rational Or Irrational?

The square root of 52 is irrational. A whole number has a rational square root only when it is a perfect square, and $52$ is not one.

Proof sketch, by contradiction:

1. Since $\sqrt{52} = 2\sqrt{13}$, if $\sqrt{52}$ were rational then $\sqrt{13} = \frac{\sqrt{52}}{2}$ would be rational too.

2. Write $\sqrt{13} = \frac{p}{q}$ in lowest terms, so $p^2 = 13q^2$.

3. Then $13$ divides $p^2$, and because $13$ is a prime number, $13$ divides $p$. Write $p = 13k$.

4. Substituting, $169k^2 = 13q^2$, so $q^2 = 13k^2$ and $13$ divides $q$ as well.

5. Now $13$ divides both $p$ and $q$, which contradicts "lowest terms". So $\sqrt{13}$, and therefore $\sqrt{52}$, is irrational.

The same argument, with $3$ in place of $13$, appears in prove that root 3 is irrational.

Is 52 A Perfect Square?

No, 52 is not a perfect square, and its last digit tells you so before any calculation. Squares of whole numbers can only end in $0, 1, 4, 5, 6$ or $9$, so a number ending in $2$, $3$, $7$ or $8$ is never a perfect square.

The bottom row never shows a $2$, so $52$ is ruled out at a glance. The test only works one way: a number ending in $1$ or $4$ can still fail to be a square (like $51$ or $54$), so for those you check the nearest squares, here $49$ and $64$, from square 1 to 50.

How Is The Square Root Of 52 Written With Exponents?

The square root of 52 can be written as $52^{ rac{1}{2}}$, because raising to the power $rac{1}{2}$ undoes squaring. The exponent rules then give the simplified form a second way:

$$52^{\frac{1}{2}} = \left(2^2 \times 13\right)^{\frac{1}{2}}$$

$$= 2^{2 \times \frac{1}{2}} \times 13^{\frac{1}{2}}$$

$$= 2 \times 13^{\frac{1}{2}} = 2\sqrt{13}$$

This exponent view is what later lets you handle expressions like $52^{\frac{3}{2}} = 52\sqrt{52} = 104\sqrt{13} \approx 374.98$; the rules behind it are in fractional exponents.

How Do You Find The Square Root Of 52?

You find the square root of 52 exactly by prime factorization and to any number of decimal places by long division. Estimation and the Babylonian method give fast approximations, and a calculator confirms the result.

Method 1: Prime Factorization

Break $52$ into primes, then pair equal primes. Each pair leaves the root as one copy.

$$52 = 2 \times 26$$

$$= 2 \times 2 \times 13$$

$$= 2^2 \times 13$$

$$\sqrt{52} = 2\sqrt{13}$$

The pair of $2$s comes out as a single $2$; the unpaired $13$ stays under the root.

Method 2: Long Division

The long-division method finds one decimal digit at a time. Pair the digits from the decimal point ($52.,00,00,00,00$), then at each step double the root found so far, write it as the start of a new divisor, and choose the largest digit $d$ that keeps the product within the current value.

At step 2 the doubled root is $14$, so the divisor is $14d$ and $142 \times 2 = 284$ fits under $300$ while $143 \times 3 = 429$ does not. The sixth digit is $0$, so the value to four decimal places is $7.2111$ with no rounding up.

Method 3: Estimating Between Perfect Squares

Since $49 < 52 < 64$, the root is $7$ plus a fraction of the gap. The simple proportion gives

$$7 + \frac{52 - 49}{64 - 49} = 7 + \frac{3}{15} = 7.2$$

A sharper estimate uses the slope of $\sqrt{x}$ at $49$, which is $\frac{1}{2 \times 7}$:

$$7 + \frac{3}{14} \approx 7.2143$$

The first estimate is $0.011$ too low and the second is $0.003$ too high, both close enough to check an answer in your head.

Method 4: The Babylonian (Heron) Method

Start with a guess $x$, then replace it with the average of $x$ and $\frac{52}{x}$. Each round roughly doubles the number of correct digits.

$$x_0 = 7.2$$

$$x_1 = \frac{1}{2}\left(7.2 + \frac{52}{7.2}\right) \approx 7.211111$$

$$x_2 = \frac{1}{2}\left(x_1 + \frac{52}{x_1}\right) \approx 7.2111026$$

Two rounds already match the true value to seven decimal places. This is the same idea as Newton's method applied to $x^2 - 52 = 0$, which some textbooks name separately.

Method 5: Calculator Check

A calculator returns $\sqrt{52} = 7.211102551$. Squaring the four-place answer is a quick check: $7.2111^2 \approx 51.9999$, just under $52$, as it should be for a value cut off rather than rounded up.

Can The Square Root Of 52 Be Written As A Fraction?

No fraction equals the square root of 52 exactly, because it is irrational. Some fractions come very close, and the best ones come from its continued fraction:

So "root 52 can be written as" $2\sqrt{13}$ exactly, as $7.2111$ to four places, or as $\frac{649}{90}$ when a close fraction is useful.

Examples Of Square Root Of 52

Example 1

Simplify $3\sqrt{52}$.

$$3\sqrt{52} = 3 \times 2\sqrt{13}$$

$$= 6\sqrt{13}$$

Final answer: $6\sqrt{13} \approx 43.27$

Example 2

A student writes $\sqrt{52} = \sqrt{49} + \sqrt{3} = 7 + 1.732 = 8.732$. Check it.

Square the result: $8.732^2 \approx 76.25$, far from $52$. The answer is also bigger than $8 = \sqrt{64}$, which is impossible for a number below $64$.

The break is the first move: a square root does not split over addition, so $\sqrt{49 + 3} \neq \sqrt{49} + \sqrt{3}$. It only splits over multiplication, which is why the factor route works.

Final answer: $\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \approx 7.2111$.

Example 3

Solve $x^2 = 52$.

$$x = \pm\sqrt{52}$$

$$x = \pm 2\sqrt{13}$$

Final answer: $x \approx 7.2111$ or $x \approx -7.2111$. The equation has two solutions even though the symbol $\sqrt{52}$ names only the positive one.

Example 4

Simplify $\sqrt{52} + \sqrt{117}$.

$$\sqrt{52} = 2\sqrt{13}$$

$$\sqrt{117} = \sqrt{9 \times 13} = 3\sqrt{13}$$

$$2\sqrt{13} + 3\sqrt{13} = 5\sqrt{13}$$

Final answer: $5\sqrt{13} \approx 18.03$

The step that looks skippable here is simplifying both roots first: until each is written over $\sqrt{13}$, they do not look like like terms at all.

Example 5

What is the smallest whole number you can multiply $52$ by to get a perfect square?

$52 = 2^2 \times 13$, so the unpaired factor is $13$.

$$52 \times 13 = 676 = 26^2$$

Final answer: $13$, giving $\sqrt{676} = 26$.

Example 6

A square garden has an area of $52\ \text{m}^2$. Find its side and perimeter.

$$\text{side} = \sqrt{52} = 2\sqrt{13} \approx 7.21\ \text{m}$$

$$\text{perimeter} = 4 \times 2\sqrt{13} = 8\sqrt{13} \approx 28.84\ \text{m}$$

Final answer: side about $7.21$ m, perimeter about $28.84$ m.

What Mistakes Do Students Make With The Square Root Of 52?

Mistake 1: Halving Instead Of Rooting

Where it slips in: Mental arithmetic under time pressure, when "square root" and "half" blur together.

Don't do this: Write $\sqrt{52} = 26$.

The correct way: Check by squaring: $26^2 = 676$, not $52$, while $7.2^2 = 51.84$ lands right next to it. The rusher who skips the squaring check is the one who keeps writing 26.

Mistake 2: Pulling Out The 4 Instead Of Its Root

Where it slips in: Simplifying, right after writing $\sqrt{52} = \sqrt{4 \times 13}$.

Don't do this: Write $4\sqrt{13}$. That equals $\sqrt{208}$, about $14.42$, double the true value.

The correct way: Only the square root of the perfect-square factor comes out: $\sqrt{4} = 2$, so the answer is $2\sqrt{13}$. Students who say "four comes out as two" aloud at this step stop making the error.

Mistake 3: Rounding Too Early Or Too Late

Where it slips in: Multi-step problems such as Example 6, where $\sqrt{52}$ feeds into later steps.

Don't do this: Replace $\sqrt{52}$ with $7.2$ at the start, then multiply; $4 \times 7.2 = 28.8$ loses the second decimal of the perimeter.

The correct way: Keep $2\sqrt{13}$ exact until the final line, then round once. The second-guesser who re-rounds at every step drifts further from the true $28.84$ with each pass.

Where Can You Get Help With Square Roots And Algebra?

Simplifying radicals gets easier fast when someone checks your factor pairs as you go. An algebra tutor can work through radicals and irrational numbers with you, and help with algebra is there for a single stuck topic.

For a steady routine, a middle school math tutor builds the perfect-square fluency this topic depends on, and math classes online give weekly practice with feedback.

Conclusion

• The square root of 52 is $2\sqrt{13}$ exactly and about $7.2111$ as a decimal.

• It is irrational because $52 = 2^2 \times 13$ is not a perfect square, and $\sqrt{13}$ is irrational.

• Prime factorization gives the exact form; long division gives as many decimals as you need.

• Estimation ($7.2$) and two Babylonian rounds ($7.2111026$) check any answer quickly.

• The common errors are halving to 26, pulling out 4 instead of 2, and rounding before the last step.

Practice These Before Moving On

1. Simplify $5\sqrt{52}$.

2. Simplify $\sqrt{52} - \sqrt{13}$.

3. Between which two whole numbers does $\sqrt{52}$ lie, and which is it closer to?

4. Find the diagonal of a rectangle with sides $4$ cm and $6$ cm, in exact form.

5. Solve $(x - 1)^2 = 52$.

Answers:

1. $10\sqrt{13} \approx 36.06$

2. $2\sqrt{13} - \sqrt{13} = \sqrt{13} \approx 3.6056$

3. Between $7$ and $8$, closer to $7$

4. $2\sqrt{13}$ cm

5. $x = 1 \pm 2\sqrt{13}$, so $x \approx 8.2111$ or $x \approx -6.2111$

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