Square Root of 34: Value 5.8310 and 5 Methods

What Is A Square Root?

A square root of a number $n$ is a number $r$ with $r \times r = n$. So the square root of 34 is the number that, multiplied by itself, gives 34; the basics are laid out in squares and square roots.

Every positive number has two square roots, one positive and one negative: $5.8310 \times 5.8310 \approx 34$ and $(-5.8310) \times (-5.8310) \approx 34$. The radical symbol $\sqrt{34}$ always means the positive one, called the principal square root, while the equation $x^2 = 34$ has both answers, $x = \pm\sqrt{34}$.

Where Does $\sqrt{34}$ Show Up And Why Does It Matter?

The cleanest home for $\sqrt{34}$ is the diagonal of a 3 by 5 rectangle. By the Pythagoras theorem, the diagonal is $\sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}$.

The same number appears as the straight-line distance from $(0, 0)$ to $(3, 5)$ on a grid, which is exactly what the distance formula computes. It is also the size of the complex number $3 + 5i$, because the modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$.

There is a quieter appearance too. 34 is itself one of the Fibonacci numbers, and it equals $3^2 + 5^2$, the squares of the consecutive Fibonacci numbers 3 and 5. That is the identity $F_{2n-1} = F_n^2 + F_{n-1}^2$ with $n = 5$, so the 3 by 5 rectangle above is a Fibonacci rectangle.

Quick Reference Table

These are the square roots of the whole numbers around 34, so you can see where $\sqrt{34}$ sits and which neighbours simplify.

Notice that $\sqrt{34}$ sits much closer to 6 than to 5. That happens because 34 is only 2 below 36 but 9 above 25.

Is 34 A Perfect Square?

No, 34 is not a perfect square. A perfect square is a whole number times itself, and the perfect squares near 34 are $25 = 5^2$ and $36 = 6^2$, with nothing in between.

Since $25 < 34 < 36$, we get $5 < \sqrt{34} < 6$. No whole number squares to 34, so its square root has a decimal part that never stops.

Is The Square Root Of 34 Rational Or Irrational?

The square root of 34 is irrational: it cannot be written as a fraction $\frac{p}{q}$ of two integers. Here is the short proof, the same pattern used to prove that root 3 is irrational.

Suppose $\sqrt{34} = \frac{p}{q}$ with $p$ and $q$ whole numbers sharing no common factor. Squaring gives:

$$p^2 = 34q^2$$

The right side is even, so $p^2$ is even, which forces $p$ to be even. Write $p = 2k$:

$$4k^2 = 34q^2$$

$$2k^2 = 17q^2$$

Now the left side is even, so $17q^2$ is even, and since 17 is odd, $q$ must be even. Both $p$ and $q$ are even, which contradicts "no common factor", so $\sqrt{34}$ is an irrational number.

How Do You Find The Square Root Of 34?

You can find the square root of 34 five ways: prime factorization tells you the exact form, and long division, estimation, Heron's method or a calculator give you the decimal. All five agree on $5.8310$.

Method 1: Prime Factorization

Prime factorization shows whether a square root can be simplified. Split 34 into primes:

$$34 = 2 \times 17$$

So $\sqrt{34} = \sqrt{2 \times 17}$. A factor leaves the radical only when it appears as a pair, and here neither 2 nor 17 has a partner, so the simplest form stays $\sqrt{34}$.

Method 2: Long Division Method

The long division method gives one exact digit at a time and never needs correcting. Write 34 as $34.00\ 00\ 00\ 00$ and group the digits in pairs from the decimal point.

1. Pair 34. The largest square not above 34 is $5^2 = 25$. Write 5 in the answer, subtract to get $34 - 25 = 9$.

2. Bring down 00. The new dividend is 900. Double the answer so far ($5 \times 2 = 10$) and look for a digit $d$ with $(100 + d) \times d \leq 900$. $108 \times 8 = 864$ works and $109 \times 9 = 981$ is too big, so the next digit is 8. Remainder $900 - 864 = 36$.

3. Bring down 00. Dividend 3600. Double 58 to get 116, and test $(1160 + d) \times d$. $1163 \times 3 = 3489$ fits, $1164 \times 4 = 4656$ does not, so the digit is 3. Remainder $3600 - 3489 = 111$.

4. Bring down 00. Dividend 11100. Double 583 to get 1166, and test $(11660 + d) \times d$. Even $11661 \times 1 = 11661$ is too big, so the digit is 0. Remainder stays 11100.

5. Bring down 00. Dividend 1110000. Double 5830 to get 11660, and test $(116600 + d) \times d$. $116609 \times 9 = 1049481$ fits, so the digit is 9. Remainder $1110000 - 1049481 = 60519$.

The digits read $5.8309\ldots$, and the next digit is 5, so $\sqrt{34} \approx 5.8310$ to 4 decimal places.

Method 3: Estimation Between 5 And 6

Estimation gets you close without a pencil-heavy layout. Start from the nearer perfect square, 36, and use the shortcut $\sqrt{n} \approx a + \frac{n - a^2}{2a}$ with $a = 6$:

$$\sqrt{34} \approx 6 + \frac{34 - 36}{12} = 6 - \frac{1}{6} \approx 5.8333$$

That is off by only about $0.0024$. A first try that splits the gap evenly between 5 and 6, $5 + \frac{9}{11} \approx 5.8182$, is worse, because the square-root curve bends; using the nearer square wins. (The square root tricks page collects more shortcuts like this.)

Method 4: Heron's (Babylonian) Method

Heron's method improves any guess $x$ with the rule $x_{\text{new}} = \frac{1}{2}\left(x + \frac{34}{x}\right)$. Starting from 6:

$$x_1 = \frac{1}{2}\left(6 + \frac{34}{6}\right) = \frac{35}{6} \approx 5.8333$$

$$x_2 = \frac{1}{2}\left(\frac{35}{6} + \frac{204}{35}\right) = \frac{2449}{420} \approx 5.8309524$$

The true value is $5.8309519\ldots$, so two steps already agree to six decimal places. This is the Newton-Raphson method applied to $x^2 - 34 = 0$, which you will meet properly in calculus.

Method 5: Calculator Check

Type 34 and press the $\sqrt{\ }$ key: the screen shows $5.830951895$. Square your rounded answer to check it, since $5.8310^2 = 34.00056$, which is as close to 34 as a 4-place decimal can get.

Can The Square Root Of 34 Be Simplified?

No. $\sqrt{34}$ is already in simplest radical form, because 34 has no perfect-square factor except 1.

You can write $\sqrt{34} = \sqrt{2} \times \sqrt{17}$, which is true, but that is not simpler, only longer. The rules for when a root does shrink are in simplifying radical expressions.

In exponent form, "root 34 can be written as" $34^{\frac{1}{2}}$, since a square root is the power one half; see rational exponents. Every one of these forms, $\sqrt{34}$, $34^{\frac{1}{2}}$ and $\sqrt{2} \times \sqrt{17}$, names the same number.

Can You Write $\sqrt{34}$ As A Fraction?

Not exactly, since an irrational number has no exact fraction form. But some fractions come remarkably close, and they come from the continued fraction of $\sqrt{34}$, which repeats the pattern $5;\ 1, 4, 1, 10$ forever.

The fraction $\frac{35}{6}$ hides a small surprise: $35^2 - 34 \times 6^2 = 1225 - 1224 = 1$. Pairs like (35, 6) solve the Pell equation $x^2 - 34y^2 = 1$, and they are exactly why $\frac{35}{6}$ is such a good stand-in for $\sqrt{34}$; it is also the number Heron's method reached after one step.

Examples Of Square Root Of 34

Example 1

Simplify $3\sqrt{34} \times \sqrt{34}$.

$$3\sqrt{34} \times \sqrt{34} = 3 \times 34$$

$$= 102$$

Final answer: 102

Example 2

Simplify $\sqrt{34}$, given that $34 = 25 + 9$.

Wrong attempt: since $34 = 25 + 9$, write $\sqrt{34} = \sqrt{25} + \sqrt{9} = 5 + 3 = 8$.

Check it: $8^2 = 64$, nowhere near 34. A square root does not split across a plus sign, because $(5 + 3)^2 = 25 + 2 \times 5 \times 3 + 9$ has a middle term the shortcut ignores.

Correct approach: $\sqrt{25 + 9}$ is simply $\sqrt{34}$. It stays $\sqrt{34} \approx 5.8310$, and the split $25 + 9$ is only useful as a picture, the 3 by 5 rectangle.

Final answer: $\sqrt{34}$, about $5.8310$

Example 3

Solve $x^2 - 34 = 0$.

$$x^2 = 34$$

$$x = \pm\sqrt{34}$$

$$x \approx 5.8310 \text{ or } x \approx -5.8310$$

Final answer: $x = \pm\sqrt{34}$

Example 4

A square garden has an area of $34,\text{m}^2$. Find its side and the length of fencing needed to enclose it.

$$\text{side} = \sqrt{34} \approx 5.83,\text{m}$$

$$\text{fencing} = 4\sqrt{34} \approx 23.32,\text{m}$$

Final answer: side about $5.83,\text{m}$, fencing about $23.32,\text{m}$

Example 5

Find the distance between $A(1, 2)$ and $B(4, 7)$.

$$AB = \sqrt{(4 - 1)^2 + (7 - 2)^2}$$

$$= \sqrt{9 + 25}$$

$$= \sqrt{34} \approx 5.83$$

This is the same 3 by 5 rectangle again, just moved away from the origin; the distance between two points always reduces to a right triangle like this.

Final answer: $\sqrt{34}$ units, about 5.83

Example 6

Given $\sqrt{34} \approx 5.8310$, find $\sqrt{0.34}$ and $\sqrt{3400}$.

$$\sqrt{0.34} = \sqrt{\frac{34}{100}} = \frac{\sqrt{34}}{10} \approx 0.5831$$

$$\sqrt{3400} = \sqrt{34 \times 100} = 10\sqrt{34} \approx 58.31$$

Students who learned long division by pairing digits handle this in seconds, because they already think of 100 as one "pair" of places; a decimal shift of two places in the number is one place in the root.

Final answer: $\sqrt{0.34} \approx 0.5831$ and $\sqrt{3400} \approx 58.31$

What Mistakes Do Students Make With $\sqrt{34}$?

Mistake 1: Writing $\sqrt{34} = \pm 5.831$

Where it slips in: Right after solving $x^2 = 34$, when the $\pm$ from the equation gets copied onto the symbol.

Don't do this: Write $\sqrt{34} = \pm 5.831$ as a single value.

The correct way: $\sqrt{34}$ is the principal root, about $+5.831$; the equation $x^2 = 34$ has two solutions, $x = \pm\sqrt{34}$, and the $\pm$ belongs to $x$, not to the radical.

Mistake 2: Turning $\sqrt{34}$ Into $2\sqrt{17}$

Where it slips in: Students who have just simplified $\sqrt{68} = 2\sqrt{17}$ see the 17 again in $34 = 2 \times 17$ and pull a 2 out by habit.

Don't do this: Write $\sqrt{34} = 2\sqrt{17}$. Squaring it gives $4 \times 17 = 68$, not 34.

The correct way: A factor leaves the root only as a perfect square. 34 contains a single 2, not $2^2 = 4$, so nothing comes out; the memorizer who learned "pull out the 2" without the pair rule is the one who trips here.

Mistake 3: Moving The Decimal By One Place

Where it slips in: Finding $\sqrt{3.4}$ from $\sqrt{34}$, where the rusher shifts the point once and writes $0.5831$.

Don't do this: Assume $\sqrt{3.4} = 0.5831$. Check it: $0.5831^2 \approx 0.34$, not 3.4.

The correct way: Digits move in pairs. $3.4 = \frac{340}{100}$, so $\sqrt{3.4} = \frac{\sqrt{340}}{10} \approx 1.8439$; the grouping step in long division feels skippable, but it is exactly what keeps the decimal point honest.

Where Can You Get Help With Square Roots And Algebra?

Square roots feed straight into quadratics, surds and the distance formula, so a shaky start here shows up again in Grade 8 and beyond. If you want a teacher to work through roots and radicals with you, an algebra tutor or a middle school math tutor can pace it to you.

For steady weekly practice, help with algebra and live math classes online cover square roots alongside the topics that build on them.

Conclusion

• The square root of 34 is $\sqrt{34} \approx 5.8310$, and the symbol always means the positive root.

• 34 is not a perfect square ($25 < 34 < 36$), so $\sqrt{34}$ is irrational and lies between 5 and 6, much nearer 6.

• $34 = 2 \times 17$ has no square factor, so $\sqrt{34}$ is already in simplest radical form.

• Long division, estimation from 36, Heron's method and a calculator all agree on $5.8310$.

• $\sqrt{34}$ is the diagonal of a 3 by 5 rectangle, the distance from $(0, 0)$ to $(3, 5)$.

Practice These Before Moving On

1. Between which two consecutive whole numbers does $\sqrt{34}$ lie?

2. Simplify $\sqrt{34} \times \sqrt{2}$.

3. Given $\sqrt{34} \approx 5.8310$, find $\sqrt{0.0034}$.

4. Solve $2x^2 = 68$.

5. Apply one step of Heron's method to the guess $5.8$ and compare it with $5.8310$.

Answers

1. Between 5 and 6, because $25 < 34 < 36$.

2. $\sqrt{68} = 2\sqrt{17} \approx 8.2462$.

3. $\sqrt{0.0034} = \frac{\sqrt{34}}{100} \approx 0.05831$.

4. $x^2 = 34$, so $x = \pm\sqrt{34} \approx \pm 5.8310$.

5. $\frac{1}{2}\left(5.8 + \frac{34}{5.8}\right) \approx 5.8310$, already correct to 4 decimal places.

If question 3 came out as $0.5831$, go back to Mistake 3 and redo the pairing.

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Read More

• Square root 1 to 30

• Square root of 10

• Square root of 25

• Perfect cube

• Diagonal of a rectangle