Square Root Of 29: Value 5.3852, Methods And Examples

What Is A Square Root?

A square root of a number $n$ is a number that gives $n$ when multiplied by itself. For 29, that means a number $r$ with $r \times r = 29$.

Every positive number has two square roots, one positive and one negative. For 29 they are $+5.3852\ldots$ and $-5.3852\ldots$, because a negative times a negative is also positive.

The symbol $\sqrt{29}$ names only the positive one, called the principal square root. So $\sqrt{29} \approx 5.3852$, while the equation $x^2 = 29$ has the two solutions $x = \pm\sqrt{29}$.

The number under the radical sign is the radicand, which here is 29. In exponent form the same value is written $29^{\frac{1}{2}}$.

Where Does The Square Root Of 29 Show Up, And Why Does It Matter?

The square root of 29 is the length of the diagonal of a 2 by 5 rectangle. By the Pythagoras theorem, that diagonal is $\sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}$.

The same number appears on a coordinate grid whenever two points are 2 units apart in one direction and 5 units apart in the other, which is how the distance between two points formula produces it. It also measures the space diagonal of a 2 by 3 by 4 box, since $2^2 + 3^2 + 4^2 = 29$.

• In number theory, $\sqrt{29}$ is tied to the Pell equation $x^2 - 29y^2 = 1$, whose smallest whole-number solution is $x = 9801$, $y = 1820$ (listed on Wolfram MathWorld).

• In estimation, the fraction $\frac{70}{13} = 5.3846\ldots$ is within 0.0006 of $\sqrt{29}$, because $70^2 - 29 \times 13^2 = -1$.

Quick Reference Table

These are the square roots from 24 to 36, the band that surrounds 29, with the simplified radical for each.

Only 25 and 36 in this band are perfect squares, so every root between them is an irrational decimal. Notice that $\sqrt{29}$ sits just under the halfway mark, 5.5, because 29 is 4 away from 25 but 7 away from 36.

Is The Square Root Of 29 Rational Or Irrational?

The square root of 29 is irrational. It cannot be written as a fraction $\frac{p}{q}$ of two whole numbers, and its decimal never ends or settles into a repeating block.

A whole number has a rational square root only when it is a perfect square. Since 29 falls between 25 and 36, it is not one, so $\sqrt{29}$ is an irrational number.

Here is the proof sketch, by contradiction. Suppose $\sqrt{29} = \frac{p}{q}$ with $p$ and $q$ whole numbers sharing no common factor.

$$29 = \frac{p^2}{q^2}$$

$$p^2 = 29q^2$$

So 29 divides $p^2$, and because 29 is prime, 29 divides $p$. Write $p = 29k$.

$$(29k)^2 = 29q^2$$

$$29k^2 = q^2$$

Now 29 divides $q^2$, so 29 divides $q$ as well. Both $p$ and $q$ share the factor 29, which breaks the "no common factor" assumption, so $\sqrt{29}$ cannot be a fraction.

This is the same argument used to prove that root 7 is irrational, and it works for the square root of any prime.

Can The Square Root Of 29 Be Simplified?

No. The square root of 29 is already in its simplest radical form, $\sqrt{29}$.

Simplifying radical expressions means pulling a perfect-square factor out from under the root, as in $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$. The only factors of 29 are 1 and 29, so there is no square factor to pull out.

The radical form does show up inside bigger roots, though. For example, $\sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29}$ and $\sqrt{261} = \sqrt{9 \times 29} = 3\sqrt{29}$.

How Do You Find The Square Root Of 29?

You can find the square root of 29 by prime factorization, long division, estimation, repeated subtraction, Newton's (Babylonian) iteration or a continued-fraction approximation. Long division gives the digits one at a time; Newton's method gets there fastest.

Method 1: Prime Factorization

Prime factorization writes the number as a product of primes and pairs them up. Each pair of equal primes comes out of the root as a single factor.

$$29 = 29 \times 1$$

Because 29 is one of the prime numbers, there is no pair to take out. Prime factorization therefore tells you the root is $\sqrt{29}$ in simplest form, but it cannot give you the decimal.

Method 2: Long Division

The long division (digit-pair) method finds one decimal digit of $\sqrt{29}$ per step. Write 29 as $29.00\ 00\ 00\ 00\ 00$, grouping the zeros in pairs after the decimal point.

1. Find the largest whole number whose square is at most 29. That is 5, since $5 \times 5 = 25$. Subtract to leave 4.

2. Bring down the next pair to make 400. Double the quotient so far (5 becomes 10) and find the digit $d$ with $(100 + d) \times d \leq 400$. Here $d = 3$, since $103 \times 3 = 309$ and $104 \times 4 = 416$ is too big.

3. Remainder $400 - 309 = 91$. Bring down 00 to make 9100. Double 53 to get 106, and $1068 \times 8 = 8544$ fits while $1069 \times 9 = 9621$ does not.

4. Remainder 556, bring down to 55600. Double 538 to get 1076, and $10765 \times 5 = 53825$ fits.

5. Remainder 1775, bring down to 177500. Double 5385 to get 10770, and $107701 \times 1 = 107701$ fits.

6. Remainder 69799, bring down to 6979900. Double 53851 to get 107702, and $1077026 \times 6 = 6462156$ fits.

The digits read 5.38516, so $\sqrt{29} \approx 5.3852$ to four decimal places. The remainder never reaches zero, which is the long division's way of showing the decimal never ends.

Method 3: Estimation Between Perfect Squares

Start from the perfect squares around 29: $5^2 = 25$ and $6^2 = 36$. So $\sqrt{29}$ lies between 5 and 6.

A straight-line guess gives $5 + \frac{29 - 25}{36 - 25} = 5 + \frac{4}{11} \approx 5.36$. That is close, but a little low, because the square-root curve bends.

A better quick estimate uses $\sqrt{a^2 + b} \approx a + \frac{b}{2a}$ with $a = 5$ and $b = 4$.

$$\sqrt{29} \approx 5 + \frac{4}{10} = 5.4$$

Check: $5.4^2 = 29.16$, only 0.16 too high, so the true value is just under 5.4.

Method 4: Repeated Subtraction Of Odd Numbers

Every perfect square is a sum of consecutive odd numbers ($1 + 3 + 5 + \ldots$). Subtract odd numbers from 29 in order and count the steps.

• $29 - 1 = 28$

• $28 - 3 = 25$

• $25 - 5 = 20$

• $20 - 7 = 13$

• $13 - 9 = 4$

The next odd number, 11, is larger than 4, so you stop after 5 subtractions with 4 left over. A leftover that is not zero confirms 29 is not a perfect square, and the count gives the whole-number part of the root, 5.

Method 5: Newton's (Babylonian) Iteration

Newton's method, which the Greeks knew as Heron's method, improves a guess $x$ with one rule.

$$x_{\text{new}} = \frac{1}{2}\left(x + \frac{29}{x}\right)$$

Start with $x_0 = 5$.

$$x_1 = \frac{1}{2}\left(5 + \frac{29}{5}\right) = 5.4$$

$$x_2 = \frac{1}{2}\left(5.4 + \frac{29}{5.4}\right) \approx 5.385185$$

$$x_3 \approx 5.3851648072$$

Three rounds already agree with the true value to 10 decimal places. (Calculus students later meet this exact rule as a tangent-line trick.)

Method 6: A Fraction That Gets Close

The square root of 29 has the repeating continued fraction $[5;\ \overline{2, 1, 1, 2, 10}]$. Cutting it off early gives fractions that close in on the root.

The fraction $\frac{70}{13}$ is the best one with a small denominator. You can check it by hand: $70^2 = 4900$ and $29 \times 13^2 = 4901$, just 1 apart.

Calculator Check

Type 29 and press the square-root key: the display shows 5.385164807. Square that result and you get back 29, up to rounding in the last digit.

Examples Of Square Root Of 29

Example 1

Round the square root of 29 to one, two and three decimal places.

$$\sqrt{29} = 5.38516\ldots$$

One decimal place: the next digit is 8, so round up to 5.4.

Two decimal places: the next digit is 5, so round up to 5.39.

Three decimal places: the next digit is 1, so keep 5.385.

The step students skip here is looking one digit past the place they want before rounding; reading only "5.38" and stopping gives the wrong 2 dp answer.

Example 2

Estimate the square root of 29 without a calculator.

Wrong path. 29 is between 25 and 36, so take the midpoint of 5 and 6:

$$\sqrt{29} \approx 5.5$$

Check it: $5.5^2 = 30.25$, which is more than 29. The midpoint overshoots because 29 is much nearer 25 than 36.

Correct path. Use the distance from 25 instead:

$$\sqrt{29} \approx 5 + \frac{29 - 25}{2 \times 5} = 5.4$$

Check it: $5.4^2 = 29.16$, just above 29, so $\sqrt{29}$ is slightly less than 5.4. The true value, 5.3852, confirms it.

Example 3

A square garden has an area of 29 square metres. Find the side length and the perimeter.

$$\text{side} = \sqrt{29} \approx 5.385 \text{ m}$$

$$\text{perimeter} = 4\sqrt{29} \approx 21.54 \text{ m}$$

Keep $4\sqrt{29}$ exact until the last step, then round.

Example 4

Find the distance between the points $(1, 2)$ and $(3, 7)$.

$$d = \sqrt{(3 - 1)^2 + (7 - 2)^2}$$

$$d = \sqrt{4 + 25}$$

$$d = \sqrt{29} \approx 5.385$$

Example 5

Solve $x^2 - 29 = 0$.

$$x^2 = 29$$

$$x = \pm\sqrt{29}$$

$$x \approx 5.3852 \text{ or } x \approx -5.3852$$

Both answers check, since squaring either one gives 29. This is the simplest kind of quadratic equation, one with no $x$ term.

Example 6

Simplify $\sqrt{116}$ and rationalise $\frac{3}{\sqrt{29}}$.

$$\sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29} \approx 10.7703$$

$$\frac{3}{\sqrt{29}} = \frac{3 \times \sqrt{29}}{\sqrt{29} \times \sqrt{29}} = \frac{3\sqrt{29}}{29} \approx 0.5571$$

Check: $3 \div 5.3852 \approx 0.5571$, matching the rationalised form.

Which Mistakes Do Students Make With The Square Root Of 29?

Most errors on $\sqrt{29}$ come from treating the root like ordinary arithmetic, from rushing the long division, or from mixing up the principal root with the two solutions of $x^2 = 29$.

Mistake 1: Splitting The Root Over A Sum

Where it slips in: Students who know $\sqrt{25} = 5$ and $\sqrt{4} = 2$ see that $25 + 4 = 29$.

Don't do this: Write $\sqrt{29} = \sqrt{25} + \sqrt{4} = 7$. Squaring 7 gives 49, nowhere near 29.

The correct way: A square root splits over multiplication, $\sqrt{ab} = \sqrt{a}\sqrt{b}$, but never over addition. The first instinct on $\sqrt{29}$ is almost always this split, and a quick squaring check catches it every time.

Mistake 2: Doubling Only The Last Digit In Long Division

Where it slips in: From step 3 of the long division onward, when the quotient already has two or more digits.

Don't do this: Double just the newest digit (3 becomes 6) and try divisors like $6d \times d$. The digits that follow come out wrong.

The correct way: Double the whole quotient so far, ignoring the decimal point: 53 becomes 106, then 538 becomes 1076. The rusher who skips writing the doubled number down is the one who usually loses this step, so write it beside each line before choosing the digit.

Mistake 3: Writing $\sqrt{29} = \pm 5.3852$

Where it slips in: Right after solving $x^2 = 29$, when the $\pm$ gets copied onto the symbol itself.

Don't do this: Claim the symbol $\sqrt{29}$ equals both $5.3852$ and $-5.3852$.

The correct way: The symbol $\sqrt{29}$ means the principal (positive) root, about 5.3852. The equation $x^2 = 29$ has two solutions, written $x = \pm\sqrt{29}$; the memorizer who learned "square roots are plus or minus" tends to attach the sign to the wrong thing.

The confusion here is real, and I had to slow down the first time I explained it: "29 has two square roots" and "$\sqrt{29}$ is positive" are both true. The first sentence is about the number 29, while the second is about what one symbol means.

Where Can You Get Help With Square Roots And Algebra?

Square roots like $\sqrt{29}$ lead straight into radicals, exponents and quadratic equations, so a gap here tends to show up again a year later. If your child wants a teacher to work through these steps live, an algebra tutor or a middle school math tutor can pace the long division and proof work.

For ongoing practice, there are also math classes online and dedicated help with algebra sessions.

Conclusion

• The square root of 29 is $\sqrt{29} \approx 5.3852$, the positive number whose square is 29.

• It is irrational and already in simplest radical form, because 29 is prime.

• Long division gives the digits 5.38516 one at a time, and Newton's method reaches 10 decimal places in three rounds.

• Geometrically, $\sqrt{29}$ is the diagonal of a 2 by 5 rectangle and the space diagonal of a 2 by 3 by 4 box.

• The most common error is splitting the root as $\sqrt{25} + \sqrt{4}$; a squaring check catches it.

Practice These Before Moving On

1. Round $\sqrt{29}$ to one decimal place.

2. Simplify $\sqrt{261}$.

3. Find the length of the diagonal of a rectangle with sides 2 cm and 5 cm.

4. Between which two whole numbers does $\sqrt{29}$ lie, and which one is it closer to?

5. Work out $\left(\sqrt{29} + 2\right)\left(\sqrt{29} - 2\right)$ without a calculator.

Answers:

1. 5.4

2. $3\sqrt{29} \approx 16.1555$, since $261 = 9 \times 29$

3. $\sqrt{29} \approx 5.385$ cm

4. Between 5 and 6, closer to 5 (29 is 4 from 25 but 7 from 36)

5. $29 - 4 = 25$

If question 2 trips you up, go back to the section on simplifying; if question 5 does, write out the difference of squares $a^2 - b^2$ first.

Book a free demo class

Read More

• Square root 1 to 30 table

• Squares and square roots

• Square root of 20

• Square root of 21

• Square root tricks