Square Root Of 28: Value, Simplest Radical Form And Long Division

What Is A Square Root?

A square root of a number $n$ is a number $r$ with $r^2 = n$. Every positive number has two square roots, one positive and one negative, so 28 has $2\sqrt{7}$ and $-2\sqrt{7}$.

The radical sign $\sqrt{\phantom{0}}$ always means the principal square root, the non-negative one. So $\sqrt{28}$ names exactly one number, $5.2915\ldots$, and the number under the sign (here 28) is called the radicand.

Where Does The Square Root Of 28 Appear, And Why Does It Matter?

The square root of 28 shows up whenever a length is forced to have a square equal to 28. Here are the places it turns up most often:

• A square with area 28: a square plot of 28 square metres has sides of $\sqrt{28} \approx 5.29$ m, so its perimeter is $4\sqrt{28} = 8\sqrt{7} \approx 21.17$ m.

• A right triangle with hypotenuse 8 and one leg 6: by the Pythagoras theorem, the other leg is $\sqrt{8^2 - 6^2} = \sqrt{64 - 36} = \sqrt{28}$.

• A triangle with sides 4 and 6 and a $60^\circ$ angle between them: the law of cosines gives the third side as $\sqrt{16 + 36 - 24} = \sqrt{28} = 2\sqrt{7}$.

• A circle with area $28\pi$: its radius is $\sqrt{28}$, because $\pi r^2 = 28\pi$ means $r^2 = 28$.

One fact you will not find on most pages: no two points on a whole-number grid are exactly $\sqrt{28}$ apart. A grid distance needs $a^2 + b^2 = 28$, and the sum of two squares theorem rules that out because 28 contains the prime 7 (which leaves remainder 3 when divided by 4) exactly once.

Quick Reference Table

The table lists the square roots of every whole number from 25 to 36, so you can see where $\sqrt{28}$ sits among its neighbours. Only 25 and 36 are perfect squares; the rest are irrational, and a few of them simplify.

Each step of 1 in $n$ adds roughly $0.09$ to $0.10$ to the root in this stretch. That steady spacing is what makes a quick mental estimate of $\sqrt{28}$ possible later on.

Is The Square Root Of 28 Rational Or Irrational?

The square root of 28 is irrational. A whole number has a rational square root only when it is one of the perfect squares, and 28 is not: it sits between $5^2 = 25$ and $6^2 = 36$.

Here is a short proof sketch. Since $\sqrt{28} = 2\sqrt{7}$, the question reduces to whether $\sqrt{7}$ is rational.

Suppose $\sqrt{7} = \frac{p}{q}$ in lowest terms. Squaring gives

$$p^2 = 7q^2$$

So 7 divides $p^2$, and because 7 is prime it divides $p$ itself. Write $p = 7k$; then $49k^2 = 7q^2$, which gives $q^2 = 7k^2$, so 7 divides $q$ as well.

Both $p$ and $q$ are now multiples of 7, which contradicts "lowest terms". The full argument is laid out step by step in prove that root 7 is irrational.

If $2\sqrt{7}$ were rational, dividing by 2 would make $\sqrt{7}$ rational too. So $\sqrt{28}$ is an irrational number, and every decimal you write for it is an approximation.

How Do You Simplify The Square Root Of 28?

You simplify the square root of 28 by pulling out its largest perfect-square factor, which is 4. The result, $2\sqrt{7}$, is the simplest radical form.

The cleanest route is prime factorization, because it shows every square factor at once:

$$28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7$$

Pair the identical primes. The pair of 2s leaves the radical as a single 2, and the unpaired 7 stays inside:

$$\sqrt{28} = \sqrt{2^2 \times 7}$$

$$\sqrt{28} = \sqrt{2^2} \times \sqrt{7}$$

$$\sqrt{28} = 2\sqrt{7}$$

Check: $(2\sqrt{7})^2 = 4 \times 7 = 28$.

Since 7 is one of the prime numbers and has no square factor, nothing more can come out, which is what "simplest radical form" means. The general rule behind this step is covered in simplifying radical expressions.

How Do You Find The Square Root Of 28?

Four methods give the decimal value, and they all land on $5.2915$. Long division gives guaranteed digits by hand, estimation gives a fast answer, the Babylonian method gives many digits in very few steps, and a calculator confirms the result.

How Do You Find The Square Root Of 28 By Long Division?

The long division method produces one correct digit of the square root per round. It borrows the bring-down rhythm of ordinary long division, with one twist: the trial divisor is always "double the answer so far, then a new digit".

Write 28 as one pair and add pairs of zeros after the decimal point: $28.,00,00,00,00,00$. Then work round by round:

How each round works:

1. The largest square not above 28 is $5^2 = 25$. Write 5 in the root and subtract to leave 3.

2. Bring down 00 to get 300. Double the root (5 becomes 10) and find the largest digit $d$ with $(10d) \times d \leq 300$: $102 \times 2 = 204$ fits, but $103 \times 3 = 309$ does not.

3. Bring down 00 to get 9600. Double 52 to get 104; $1049 \times 9 = 9441$ fits, so the digit is 9.

4. Bring down 00 to get 15900. Double 529 to get 1058; $10581 \times 1 = 10581$ fits, but $10582 \times 2 = 21164$ is too big.

5. Bring down 00 to get 531900. Double 5291 to get 10582; $105825 \times 5 = 529125$ fits, but $105826 \times 6 = 634956$ is too big.

6. Bring down 00 to get 277500. Double 52915 to get 105830; even $1058301 \times 1$ is too big, so the digit is 0.

So $\sqrt{28} = 5.29150\ldots$, and the remainder never reaches zero, which is exactly what irrational means. To 4 decimal places, $\sqrt{28} \approx 5.2915$.

Between Which Two Numbers Does The Square Root Of 28 Lie?

The square root of 28 lies between 5 and 6, because $5^2 = 25 < 28 < 36 = 6^2$. It is much closer to 5, since 28 is only 3 above 25 but 8 below 36.

Two quick estimates tighten that range:

• Linear estimate: 28 is 3 of the 11 steps from 25 to 36, so $\sqrt{28} \approx 5 + \frac{3}{11} \approx 5.27$. That is about $0.02$ too low, because the square-root curve bends.

• Averaging estimate: divide 28 by the first guess 5 to get $5.6$, then average: $\frac{5 + 5.6}{2} = 5.3$. That is only about $0.009$ too high.

A squaring check shows which way to adjust: $5.3^2 = 28.09$ is just above 28, and $5.29^2 = 27.9841$ is just below it. So $5.29 < \sqrt{28} < 5.3$.

How Does The Babylonian Method Find The Square Root Of 28?

The Babylonian method (also called Heron's method) improves any guess $x$ with one formula:

$$x_{\text{new}} = \frac{1}{2}\left(x + \frac{28}{x}\right)$$

Start from $x_0 = 5$:

The number of correct digits roughly doubles each step, a property called quadratic convergence that is described in methods of computing square roots. The Newton-Raphson method applied to $f(x) = x^2 - 28$ produces exactly the same formula, so it gives the same table.

How Do You Check The Square Root Of 28 On A Calculator?

Type 28 and press the square-root key; the display shows $5.291502622$. To confirm by hand, square the rounded value: $5.2915^2 = 27.99997225$, which is within $0.00003$ of 28.

What Is The Square Root Of 28 Rounded To The Nearest Tenth And Hundredth?

The square root of 28 is $5.3$ to the nearest tenth and $5.29$ to the nearest hundredth. Both come from the long division digits $5.29150\ldots$.

Notice that 5.29 and 5.3 are both correct roundings of the same number; they answer different questions. When a problem says "exact", the answer is $2\sqrt{7}$, not any of these decimals.

Is The Square Root Of 28 Positive, Negative, Or Both?

The symbol $\sqrt{28}$ means only the positive value, $5.2915\ldots$. The negative partner, $-\sqrt{28}$, appears when you solve an equation such as $x^2 = 28$, whose two solutions are $x = \pm 2\sqrt{7}$.

Some pages write "the square root of 28 is $\pm 5.2915$", which mixes the two ideas. Textbooks genuinely phrase this loosely, which is why even careful students mix it up; the clean version is that 28 has two square roots, but the radical sign names one of them, as Wolfram MathWorld states for the principal square root.

Examples Of Square Root Of 28

Example 1

Simplify $3\sqrt{28}$ and give its decimal value.

$$3\sqrt{28} = 3 \times 2\sqrt{7}$$

$$3\sqrt{28} = 6\sqrt{7}$$

$$6\sqrt{7} \approx 6 \times 2.6458 \approx 15.8745$$

Final answer: $6\sqrt{7} \approx 15.8745$

Example 2

Simplify $\sqrt{28} + \sqrt{7}$.

Wrong path: add the numbers under the roots, $\sqrt{28} + \sqrt{7} = \sqrt{35} \approx 5.92$.

That answer breaks a basic check. $\sqrt{28}$ alone is already about $5.29$, and adding $\sqrt{7} \approx 2.65$ must give something near $7.94$, not $5.92$.

Correct path: simplify first, then combine like radicals.

$$\sqrt{28} + \sqrt{7} = 2\sqrt{7} + \sqrt{7}$$

$$\sqrt{28} + \sqrt{7} = 3\sqrt{7}$$

$$3\sqrt{7} \approx 7.9373$$

Final answer: $3\sqrt{7} \approx 7.9373$ (which also equals $\sqrt{63}$)

Example 3

Solve $x^2 = 28$.

$$x = \pm\sqrt{28}$$

$$x = \pm 2\sqrt{7}$$

$$x \approx \pm 5.2915$$

Final answer: $x = 2\sqrt{7}$ or $x = -2\sqrt{7}$

The negative solution is the step most often dropped here, because the radical sign alone only gives the positive root; the "$\pm$" has to be written in by you.

Example 4

A square garden has an area of 28 m². How long is each side, and how much fencing goes around it?

$$\text{side} = \sqrt{28} = 2\sqrt{7} \approx 5.29 \text{ m}$$

$$\text{fencing} = 4 \times 2\sqrt{7} = 8\sqrt{7} \approx 21.17 \text{ m}$$

Final answer: each side is about 5.29 m and the fence is about 21.17 m long. Here a negative length makes no sense, so only the positive root is used.

Example 5

A ladder 8 m long leans against a wall with its foot 6 m from the wall. How high up the wall does it reach?

$$h^2 + 6^2 = 8^2$$

$$h^2 = 64 - 36 = 28$$

$$h = \sqrt{28} = 2\sqrt{7} \approx 5.29 \text{ m}$$

Final answer: about 5.29 m up the wall.

Example 6

Simplify $\frac{14}{\sqrt{28}}$ and give its value to 4 decimal places.

$$\frac{14}{\sqrt{28}} = \frac{14}{2\sqrt{7}} = \frac{7}{\sqrt{7}}$$

$$\frac{7}{\sqrt{7}} = \frac{7\sqrt{7}}{7} = \sqrt{7}$$

$$\sqrt{7} \approx 2.6458$$

Final answer: $\sqrt{7} \approx 2.6458$

Where Do Students Go Wrong With The Square Root Of 28?

Mistake 1: Pulling Out A Factor That Is Not A Perfect Square

Where it slips in: Simplifying $\sqrt{28}$ after splitting 28 as $2 \times 14$.

Don't do this: Writing $\sqrt{28} = 2\sqrt{14}$. That squares to $4 \times 14 = 56$, not 28.

The correct way: Only a perfect-square factor can leave the radical: $28 = 4 \times 7$, so $\sqrt{28} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}$. The first instinct is to grab the first factor pair that comes to mind, $2 \times 14$, and pull the 2 out; factoring all the way to primes ($2 \times 2 \times 7$) makes the pair visible and stops that habit.

Mistake 2: Splitting The Square Root Across A Sum

Where it slips in: Mental estimates, where 28 is broken into "nice" pieces like $25 + 3$.

Don't do this: Writing $\sqrt{28} = \sqrt{25} + \sqrt{3} = 5 + 1.73 = 6.73$. Squaring $6.73$ gives about $45.3$, far from 28.

The correct way: A square root splits across multiplication, never across addition: $\sqrt{ab} = \sqrt{a}\sqrt{b}$, but $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. The rusher who sees $25 + 3$ and grabs $\sqrt{25}$ is usually fixed by one habit: square the answer before writing it down.

Mistake 3: Writing $\sqrt{28} = \pm 5.2915$

Where it slips in: Copying a calculator answer into a definition, or mixing up "the square root" with "the solutions of $x^2 = 28$".

Don't do this: Treating the symbol $\sqrt{28}$ as two numbers, or dropping $-2\sqrt{7}$ when solving $x^2 = 28$.

The correct way: $\sqrt{28}$ is one positive number, $2\sqrt{7}$; the equation $x^2 = 28$ has two solutions, $\pm 2\sqrt{7}$. The memorizer who learned "square root means plus or minus" applies it everywhere; tie the $\pm$ to the equation, not to the symbol.

Who Can Help Your Child Practise Square Roots Like $\sqrt{28}$?

A teacher who watches the factoring step live can spot the $2\sqrt{14}$ habit before it sets in. If your child wants that kind of guided practice, an algebra tutor or a middle school math tutor can work through radicals one factor tree at a time.

For broader support, help with algebra covers simplifying radicals alongside equations, and math classes online build the same skills in a live group setting.

Conclusion

• The square root of 28 is $2\sqrt{7} \approx 5.2915$, and it is irrational because 28 is not a perfect square.

• Prime factorization, $28 = 2^2 \times 7$, gives the simplest radical form: the pair of 2s comes out as one 2 and the 7 stays inside.

• Long division gives $5.29150\ldots$ digit by digit; the value rounds to $5.3$ (tenth) and $5.29$ (hundredth).

• The most common error is $2\sqrt{14}$; only a perfect-square factor can leave the radical.

• $\sqrt{28}$ names the positive root, while $x^2 = 28$ has two solutions, $\pm 2\sqrt{7}$.

Practice These Before Moving On

1. Simplify $\sqrt{112}$.

2. Evaluate $\sqrt{28} \times \sqrt{63}$.

3. Without a calculator, give the square root of 28 to one decimal place.

4. Solve $2x^2 = 56$.

5. Simplify $\sqrt{2800}$ and give its value to 2 decimal places.

Answer to Question 1: $\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7} \approx 10.5830$

Answer to Question 2: $\sqrt{28 \times 63} = \sqrt{1764} = 42$

Answer to Question 3: $5.3$, since $5.3^2 = 28.09$ and $5.2^2 = 27.04$, and 28 is much closer to $28.09$

Answer to Question 4: $x^2 = 28$, so $x = \pm 2\sqrt{7} \approx \pm 5.2915$

Answer to Question 5: $\sqrt{2800} = \sqrt{28} \times \sqrt{100} = 20\sqrt{7} \approx 52.92$

If Question 1 felt tricky, go back to the factor tree in the simplifying section and look for the largest perfect-square factor first.

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