Square Root Of 48: Simplified As 4√3, Value And Methods

What Is A Square Root?

A square root of $n$ is a number that gives $n$ when multiplied by itself. The square root of 48 is the number $r$ with $r \times r = 48$.

Every positive number has a positive and a negative square root. The symbol $\sqrt{48}$ means the positive one, the principal square root, about $6.9282$; the equation $x^2 = 48$ has both answers, $x = \pm 4\sqrt{3}$.

Where Does $\sqrt{48}$ Show Up, And Why Does It Matter?

The height of an equilateral triangle with side 8 is exactly $\sqrt{48}$. Halving the base gives a right triangle with hypotenuse 8 and base 4, so the height is $\sqrt{8^2 - 4^2} = \sqrt{64 - 16} = \sqrt{48} = 4\sqrt{3}$; this is the height of an equilateral triangle formula $\frac{\sqrt{3}}{2}a$ with $a = 8$.

That same right triangle is a 30-60-90 triangle with sides $4$, $4\sqrt{3}$ and $8$, which is why $4\sqrt{3}$ turns up all over trigonometry. In three dimensions, the long diagonal of a cube with edge 4 is $\sqrt{4^2 + 4^2 + 4^2} = \sqrt{48}$, the diagonal of a cuboid formula with all three edges equal.

Quick Reference Table

Here are the square roots of the whole numbers around 48. Several neighbours simplify the same way 48 does, by pulling a square factor out.

$\sqrt{48}$ sits just below 7, because 48 is only 1 short of the perfect square 49.

Is 48 A Perfect Square?

No. 48 lies between the perfect squares $36 = 6^2$ and $49 = 7^2$, and no whole number squares to 48.

48 does, however, contain a perfect square: $48 = 16 \times 3$. That hidden 16 is exactly what lets $\sqrt{48}$ simplify, even though 48 itself is not a perfect square.

Is The Square Root Of 48 Rational Or Irrational?

$\sqrt{48}$ is irrational. Since $\sqrt{48} = 4\sqrt{3}$, it is a rational number (4) times the square root of 3, which is known to be irrational.

The argument is short. If $4\sqrt{3}$ were a fraction $\frac{p}{q}$, then $\sqrt{3} = \frac{p}{4q}$ would also be a fraction, which is impossible; so $\sqrt{48}$ is an irrational number whose decimal never ends or repeats.

How Do You Simplify The Square Root Of 48?

To simplify the square root of 48, find the largest perfect square that divides 48, which is 16, and take its root outside. The result is $4\sqrt{3}$, the simplest radical form.

Method 1: Prime Factorization

Write 48 as a product of primes and pair equal factors:

$$48 = 2 \times 2 \times 2 \times 2 \times 3$$

$$48 = (2 \times 2) \times (2 \times 2) \times 3$$

$$\sqrt{48} = 2 \times 2 \times \sqrt{3}$$

$$\sqrt{48} = 4\sqrt{3}$$

Each pair of 2s leaves the root as one 2, and the unpaired 3 stays inside. The general rules are collected in simplifying radical expressions.

Method 2: The Largest Square Factor

List the perfect squares that divide 48: 1, 4 and 16. Use the largest one, 16:

$$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$$

Using 4 instead of 16 also works, but it takes two rounds: $\sqrt{48} = 2\sqrt{12}$, then $\sqrt{12} = 2\sqrt{3}$, giving $2 \times 2\sqrt{3} = 4\sqrt{3}$.

How Do You Find The Square Root Of 48 By Long Division?

The long division method produces the decimal of $\sqrt{48}$ one digit at a time. Write $48.00\ 00\ 00\ 00$ and work pair by pair.

1. Pair 48. The largest square not above 48 is $6^2 = 36$. Write 6; remainder $48 - 36 = 12$.

2. Bring down 00. Dividend 1200. Double 6 to get 12 and find $d$ with $(120 + d) \times d \leq 1200$: $129 \times 9 = 1161$ fits. Remainder $39$.

3. Bring down 00. Dividend 3900. Double 69 to get 138: $1382 \times 2 = 2764$ fits, $1383 \times 3 = 4149$ does not. Remainder $1136$.

4. Bring down 00. Dividend 113600. Double 692 to get 1384: $13848 \times 8 = 110784$ fits. Remainder $2816$.

5. Bring down 00. Dividend 281600. Double 6928 to get 13856: $138562 \times 2 = 277124$ fits. Remainder $4476$.

So $\sqrt{48} = 6.9282\ldots$, which is $6.9282$ to 4 decimal places. Check it against the exact form: $4 \times 1.7320508 = 6.9282032$.

How Can You Estimate $\sqrt{48}$ Without A Calculator?

Start from the nearest perfect square, 49, and adjust: $\sqrt{48} \approx 7 - \frac{1}{2 \times 7} = 7 - \frac{1}{14} \approx 6.9286$. That is within $0.0004$ of the true value, which is why "square root of 48 is approximately equal to 6.93" is the standard two-decimal answer.

Heron's method sharpens any guess with $x_{\text{new}} = \frac{1}{2}\left(x + \frac{48}{x}\right)$. From 7 it gives $\frac{97}{14} \approx 6.928571$, and one more step gives $\frac{18817}{2716} \approx 6.92820324$, correct to seven decimal places.

There is a neat check hiding in those fractions: $97^2 - 48 \times 14^2 = 9409 - 9408 = 1$. Pairs like (97, 14) solve the Pell equation $x^2 - 48y^2 = 1$, which is why $\frac{97}{14}$ is such a good fraction for $\sqrt{48}$ (the square root tricks page has more estimation shortcuts).

Can You Write The Square Root Of 48 As A Fraction?

Not exactly, because $\sqrt{48}$ is irrational. The closest simple fractions come from its continued fraction, which repeats $6;\ 1, 12$.

In exponent form, "root 48 can be written as" $48^{\frac{1}{2}}$, and that also equals $4 \times 3^{\frac{1}{2}}$.

Examples Of Square Root Of 48

Example 1

Simplify $3\sqrt{48}$.

$$3\sqrt{48} = 3 \times 4\sqrt{3}$$

$$= 12\sqrt{3}$$

$$\approx 20.7846$$

Final answer: $12\sqrt{3}$

Example 2

Simplify $\sqrt{48} + \sqrt{3}$.

Wrong attempt: add the numbers under the roots, $\sqrt{48} + \sqrt{3} = \sqrt{51}$.

Check it: $\sqrt{51} \approx 7.1414$, but $\sqrt{48} + \sqrt{3} \approx 6.9282 + 1.7321 = 8.6603$. The two answers do not match, so adding radicands is not allowed.

Correct approach: simplify first so both terms share $\sqrt{3}$, then add the coefficients.

$$\sqrt{48} + \sqrt{3} = 4\sqrt{3} + 1\sqrt{3}$$

$$= 5\sqrt{3} \approx 8.6603$$

Final answer: $5\sqrt{3}$

Example 3

Simplify $\sqrt{48} \div \sqrt{243}$.

$$\frac{\sqrt{48}}{\sqrt{243}} = \sqrt{\frac{48}{243}}$$

$$= \sqrt{\frac{16}{81}}$$

$$= \frac{4}{9}$$

Both 48 and 243 share a factor of 3, and the square root of 243 is $9\sqrt{3}$, so the $\sqrt{3}$ cancels: $\frac{4\sqrt{3}}{9\sqrt{3}} = \frac{4}{9}$.

Final answer: $\frac{4}{9}$

Example 4

An equilateral triangle has sides of 8 cm. Find its height and area.

$$h = \sqrt{8^2 - 4^2} = \sqrt{48} = 4\sqrt{3} \approx 6.93,\text{cm}$$

$$\text{area} = \frac{1}{2} \times 8 \times 4\sqrt{3} = 16\sqrt{3} \approx 27.71,\text{cm}^2$$

Most students find the height correctly and then lose marks on the area by rounding $4\sqrt{3}$ to 6.9 too early; keeping $16\sqrt{3}$ until the last line avoids that.

Final answer: height $4\sqrt{3} \approx 6.93,\text{cm}$, area $16\sqrt{3} \approx 27.71,\text{cm}^2$

Example 5

Rationalise $\frac{12}{\sqrt{48}}$.

$$\frac{12}{\sqrt{48}} = \frac{12}{4\sqrt{3}} = \frac{3}{\sqrt{3}}$$

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

$$= \sqrt{3} \approx 1.7321$$

Final answer: $\sqrt{3}$

Example 6

Solve $(x + 1)^2 = 48$.

$$x + 1 = \pm\sqrt{48} = \pm 4\sqrt{3}$$

$$x = -1 + 4\sqrt{3} \approx 5.9282$$

$$x = -1 - 4\sqrt{3} \approx -7.9282$$

Final answer: $x = -1 \pm 4\sqrt{3}$

What Mistakes Do Students Make With $\sqrt{48}$?

Mistake 1: Stopping At $2\sqrt{12}$

Where it slips in: Spotting the factor 4 first, because $48 = 4 \times 12$ is the split most students see.

Don't do this: Write $\sqrt{48} = 2\sqrt{12}$ and call it simplified.

The correct way: Check the number left inside: 12 still contains 4, so $2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}$. Looking for the largest square factor (16) first is the habit that removes this extra round.

Mistake 2: Pulling Out 16 Instead Of 4

Where it slips in: Splitting $48 = 16 \times 3$ correctly, then moving the 16 itself outside the root.

Don't do this: Write $\sqrt{48} = 16\sqrt{3}$. Check: $16\sqrt{3} \approx 27.7$, far bigger than $\sqrt{49} = 7$.

The correct way: Only the root of the square factor comes out: $\sqrt{16} = 4$, so $\sqrt{48} = 4\sqrt{3}$. The rusher who skips the "is this smaller than 7?" check never notices the answer is impossible.

Mistake 3: Rounding $\sqrt{3}$ Too Early

Where it slips in: Turning $4\sqrt{3}$ into a decimal with $\sqrt{3} \approx 1.73$.

Don't do this: Write $\sqrt{48} \approx 4 \times 1.73 = 6.92$ and report 6.92 as the 2-decimal value.

The correct way: Use enough digits of $\sqrt{3}$ for the final accuracy: $4 \times 1.7321 = 6.9284$, which still rounds to 6.93, while the true value is 6.9282. The silent understander gets the exact form right and then loses accuracy at this last, seemingly harmless step.

Where Can You Get Help With Simplifying Square Roots?

Simplifying $\sqrt{48}$ to $4\sqrt{3}$ is the skill behind surds, the quadratic formula and most trigonometry exact values, so it is worth making automatic. An algebra tutor can drill the largest-square-factor habit until it sticks.

For regular structured practice, help with algebra and math classes online carry radicals forward into equations and geometry.

Conclusion

• The square root of 48 is $4\sqrt{3}$, about $6.9282$.

• 48 is not a perfect square, but it contains the square factor 16, which is why $\sqrt{48}$ simplifies.

• $\sqrt{48}$ is irrational because it is 4 times the irrational $\sqrt{3}$.

• Long division confirms $6.9282$, and $7 - \frac{1}{14}$ gives a quick estimate.

• Simplify radicals before adding them: $\sqrt{48} + \sqrt{3} = 5\sqrt{3}$, not $\sqrt{51}$.

Practice These Before Moving On

1. Simplify $\sqrt{48} \times \sqrt{3}$.

2. Simplify $\sqrt{4800}$.

3. Simplify $\sqrt{48} \div \sqrt{4}$.

4. Solve $x^2 = 48$.

5. Find the long diagonal of a cube with edge 4 cm.

Answers

1. $\sqrt{144} = 12$.

2. $\sqrt{48 \times 100} = 10 \times 4\sqrt{3} = 40\sqrt{3} \approx 69.2820$.

3. $\sqrt{12} = 2\sqrt{3} \approx 3.4641$.

4. $x = \pm 4\sqrt{3} \approx \pm 6.9282$.

5. $\sqrt{4^2 + 4^2 + 4^2} = \sqrt{48} = 4\sqrt{3} \approx 6.93$ cm.

If question 2 came out as $4\sqrt{30}$, look again at how 100 leaves the root as 10.

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