Choose "Simplify one radical" or "Two radicals", then enter the number in front of the radical and the number under it. The formula below is linked to the input fields, so you can also edit the numbers in the formula directly.
Enter numbers using digits. A blank coefficient counts as 1, and it can be a decimal, a negative number or a fraction such as 3/4. The number under the radical must be a whole number of 1 or more.
Result and graph
Enter the number under the radical in the fields on the left and press "Calculate". The simplified form and a number line will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and figures
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Simplify a square root such as \(\sqrt{48}\) to \(4\sqrt{3}\) (the form with the smallest possible whole number under the radical). The steps show the prime factorization of the radicand and how the perfect square comes out
- You can also see that when the radicand is a perfect square, the radical goes away and you get a whole number, as in \(\sqrt{36} = 6\)
- Add, subtract, multiply and divide radicals such as \(3\sqrt{2} + 5\sqrt{2}\) and \(2\sqrt{6} \times 3\sqrt{2}\), with exact answers in simplest form
- When a division leaves a radical in the denominator, the answer is given with the denominator rationalized. Example: \(6 \div \sqrt{3} = 2\sqrt{3}\)
- The exact form is the main answer, along with its decimal value and the two whole numbers it lies between. A number line shows its size too
- The calculator shows the radical just as you would write it by hand, and you can edit the numbers inside or in front of it directly
This page handles square roots of positive whole numbers. It does not handle square roots of negative numbers, cube roots or other higher roots, or nested radicals such as \(\sqrt{5 + 2\sqrt{6}}\).
What is this calculation used for?
The f-stops on a camera lens (photography)
The f-numbers on a camera lens go 1.4, 2, 2.8, 4, 5.6, 8, 11, 16. Each step lets in half as much light, which means the area of the lens opening halves, so its diameter shrinks by a factor of \(\sqrt{2}\). That is why each f-number is about \(\sqrt{2} \approx 1.414\) times the one before.
In fact, f/1.4 is \(\sqrt{2}\), f/2.8 is \(2\sqrt{2} \approx 2.83\), f/5.6 is \(4\sqrt{2} \approx 5.66\) and f/11 is \(8\sqrt{2} \approx 11.3\), rounded for the lens markings. Simplified radicals show the pattern hidden in the numbers.
Finding a diagonal length (building, design, surveying)
When you find a diagonal from two sides that meet at a right angle (the Pythagorean theorem), the answer is often a radical. The diagonal of a square 1 ft on each side is \(\sqrt{2} \approx 1.414\) ft, and for sides of 3 ft and 4 ft it is \(\sqrt{25} = 5\) ft.
Carpenters and builders measure this diagonal to check that a corner is square (the 3-4-5 rule). On a plan, instead of writing a value like \(\sqrt{18}\) ft, write it simplified as \(3\sqrt{2}\) ft, and you can tell at once that it is 3 times \(\sqrt{2} \approx 1.414\), about 4.24 ft.
Standard deviation, the spread of data (statistics and quality control)
The standard deviation is the square root of the average of the squared distances from the mean (the variance). Squaring and then taking the square root brings the result back to the same units as the original data.
For example, the five values 4, 6, 8, 10, 12 have a mean of 8 and a variance of 8, so the standard deviation is \(\sqrt{8} = 2\sqrt{2} \approx 2.83\). Test scores, product variation in factories, reference ranges in medical checkups: square roots appear wherever data is handled.
How long a fall takes, and the swing of a pendulum (physics)
An object dropped from a height of \(h\) feet reaches the ground in \(\sqrt{\dfrac{2h}{g}}\) seconds (\(g\), the acceleration of gravity, is about 32.2 ft/s²). From 16 ft, that is about 1.0 second. The time for one full swing of a pendulum is also proportional to the square root of its length (about 2.0 seconds for a pendulum 1 m, or about 39 inches, long).
This rule, "time is proportional to the square root of distance", is used in everyday settings such as designing clocks and playground equipment and investigating falls.
Working out speed from skid marks (crash investigation)
From the length of the skid marks a car leaves on the road, investigators in the US estimate the speed when braking began with \(\sqrt{30df}\) mph (\(d\) is the skid length in feet and \(f\) is the drag factor between the tires and the road). On dry pavement with \(f = 0.7\) and skid marks 66 ft long, that is \(\sqrt{1386} \approx 37.2\) mph.
This estimate is really used in crash investigations. It also explains a key safe-driving fact: doubling your speed makes the braking distance (from when the brakes start working until you stop) four times as long, because braking distance is proportional to the square of the speed.
Formulas and figures
What a square root and the radical sign mean
Figure
Standard notation
(the usual math form)
\(\left(\sqrt{a}\right)^{2}\)
\(=\)
\(a\)
In words
(symbols replaced with words)
①
\(\sqrt{a}\), the positive number whose square is \(a\), squared again
\(=\)
②
\(a\): number under the radical
The formula in words
①
For a number \(a\) of 0 or more, take
\(\sqrt{a}\), the positive number whose square is \(a\), squared again
②
and you are back to the
\(a\): number under the radical
itself
Quick example
The square root of \(2\), \(\sqrt{2}\), is the length of the diagonal of a square with side \(1\). Squaring it gives
the diagonal \(\sqrt{2}\) squared
\(=\)
number under the radical, \(2\)
\(\left(\sqrt{2}\right)^{2} = \sqrt{2} \times \sqrt{2} = 2\)
\(\sqrt{2} \approx 1.41421356\)
Key idea
Picture a square with area \(a\). The length of its side is exactly \(\sqrt{a}\). Squaring the side gives back the area \(a\), and that is what the radical sign means.
\(\sqrt{2}\) and \(\sqrt{3}\) cannot be written as fractions (they are irrational numbers). As decimals they never end and never repeat. So instead of writing \(1.41421356\dots\) forever, the single symbol \(\sqrt{2}\) gives the exact value.
There are two numbers whose square is \(a\): \(+\sqrt{a}\) and \(-\sqrt{a}\). By agreement, the symbol \(\sqrt{a}\) means only the positive one (the principal square root).
Simplifying a radical (take the perfect square out)
Figure
Standard notation
(the usual math form)
\(\sqrt{a^{2}b}\)
\(=\)
\(a\)
\(\sqrt{b}\)
In words
(symbols replaced with words)
③
a square root whose radicand is a perfect square \(a^2\) times the rest \(b\)
\(=\)
①
\(a\): root of the perfect square (comes outside)
②
\(\sqrt{b}\): square root of the rest (stays inside)
The formula in words
①
For numbers \(a\) and \(b\) of 0 or more, split the radicand into "perfect square × the rest". Take the
\(a\): root of the perfect square
outside the radical
②
leave the
\(\sqrt{b}\): square root of the rest
inside, and multiply. The result equals the
③
original square root \(\sqrt{a^2b}\)
Quick example
\(48 = 2^4 \times 3 = 4^2 \times 3\), so \(\sqrt{48}\) becomes
original square root \(\sqrt{48}\)
\(=\)
\(4\) comes outside
\(\sqrt{3}\) stays inside
\(48 = 2^{4} \times 3 = 4^{2} \times 3\)
\(\sqrt{48} = \sqrt{4^{2} \times 3} = 4\sqrt{3} \approx 6.9282\)
Key idea
To see why it can come out, split it as \(\sqrt{a^2b} = \sqrt{a^2} \times \sqrt{b}\). \(\sqrt{a^2}\) is "the positive number whose square is \(a^2\)", which is just \(a\). So \(a\) alone comes outside the radical, and \(\sqrt{b}\) stays inside.
Prime factorization is the sure way to find perfect squares. Every pair of the same prime factor lets one copy of that prime come outside, and only the primes left without a partner stay inside. For example, \(72 = 2^3 \times 3^2\). There are three 2s, which make one pair, so one 2 comes out and one 2 stays in. There are two 3s, exactly one pair, so one 3 comes out. So \(\sqrt{72} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}\).
There are two reasons to simplify. First, the size is easier to see: \(\sqrt{48}\) means little at a glance, but \(4\sqrt{3}\) is 4 times \(\sqrt{3} \approx 1.73\), so about 6.9. Second, equal values end up in the same form, so you can check answers and tidy up expressions.
Adding and subtracting radicals (combine like terms)
Figure
Standard notation
(the usual math form)
\(p\sqrt{a}\)
\(+\)
\(q\sqrt{a}\)
\(=\)
\(\left(p+q\right)\sqrt{a}\)
In words
(symbols replaced with words)
①
\(p\) copies of \(\sqrt{a}\)
\(+\)
②
\(q\) copies of \(\sqrt{a}\)
\(=\)
③
\(p+q\) copies of \(\sqrt{a}\)
The formula in words
①
When the radicands are the same \(a\),
\(p\) copies of \(\sqrt{a}\)
②
plus
\(q\) copies of \(\sqrt{a}\)
are same-size pieces, so you can add the counts, which gives
③
\(p+q\) copies of \(\sqrt{a}\)
Quick example
In \(3\sqrt{2} + 5\sqrt{2}\), both are made of \(\sqrt{2}\) pieces, so
3 copies of \(\sqrt{2}\)
\(+\)
5 copies of \(\sqrt{2}\)
\(=\)
8 copies of \(\sqrt{2}\)
\(3\sqrt{2} + 5\sqrt{2} = \left(3+5\right)\sqrt{2} = 8\sqrt{2} \approx 11.3137\)
Key idea
It is the same idea as "3 apples + 5 apples = 8 apples". When you combine \(3x + 5x = 8x\) in algebra, the \(x\) here is simply \(\sqrt{2}\). This is called combining like terms.
On the other hand, radicals with different radicands cannot be combined into one. The most common mistake is to add the insides, writing \(\sqrt{a} + \sqrt{b} = \sqrt{a+b}\). This is not true. An example with whole numbers makes it clear: \(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\), but \(\sqrt{9 + 16} = \sqrt{25} = 5\), a completely different number. Subtraction is the same: \(\sqrt{a} - \sqrt{b}\) is not \(\sqrt{a-b}\) either.
So \(\sqrt{2} + \sqrt{3}\) is not \(\sqrt{5}\) (in fact \(\sqrt{2} + \sqrt{3} \approx 3.1463\), while \(\sqrt{5} \approx 2.2361\)). The answer is \(\sqrt{2} + \sqrt{3}\) as it is. Not forcing together things that cannot be combined is the correct way to answer.
Radicals that look different can become like terms after simplifying. \(\sqrt{2} + \sqrt{8}\) does not seem to combine, but \(\sqrt{8} = 2\sqrt{2}\), so \(\sqrt{2} + 2\sqrt{2} = 3\sqrt{2}\). For addition and subtraction, the first step is always to simplify both radicals.
Multiplying radicals (multiply the insides to make one)
Figure
Standard notation
(the usual math form)
\(\sqrt{a}\)
\(\times\)
\(\sqrt{b}\)
\(=\)
\(\sqrt{ab}\)
In words
(symbols replaced with words)
①
square root of \(a\)
\(\times\)
②
square root of \(b\)
\(=\)
③
square root of the product of the insides, \(ab\)
The formula in words
①
For numbers \(a\) and \(b\) that are both 0 or more, multiplying the
square root of \(a\), \(\sqrt{a}\)
②
by the
square root of \(b\), \(\sqrt{b}\)
multiplies the insides together, which gives the
③
square root of \(ab\), \(\sqrt{ab}\)
Quick example
For \(\sqrt{3} \times \sqrt{6}\), multiply the insides to get \(\sqrt{18}\), then simplify
\(\sqrt{3}\)
\(\times\)
\(\sqrt{6}\)
\(=\)
\(\sqrt{18}\) (simplified, \(3\sqrt{2}\))
\(\sqrt{3} \times \sqrt{6} = \sqrt{3 \times 6} = \sqrt{18}\)
\(\sqrt{18} = \sqrt{3^{2} \times 2} = 3\sqrt{2} \approx 4.2426\)
Key idea
It makes sense if you think of a rectangle with height \(\sqrt{a}\) and width \(\sqrt{b}\): its area is \(\sqrt{ab}\). Unlike addition, multiplication can combine radicals into one even when the radicands are different.
If there are numbers in front of the radicals, multiply the numbers in front together and the insides together separately. For example, \(2\sqrt{6} \times 3\sqrt{2} = (2 \times 3)\sqrt{6 \times 2} = 6\sqrt{12}\). Then simplify \(\sqrt{12} = 2\sqrt{3}\) to get \(6 \times 2\sqrt{3} = 12\sqrt{3}\). After multiplying, the radicand gets bigger and often contains a perfect square, so do not forget to simplify at the end.
Multiplying a radical by itself removes the radical (\(\sqrt{5} \times \sqrt{5} = \sqrt{25} = 5\)). This says the same thing as the first formula, \(\left(\sqrt{a}\right)^2 = a\).
Note that this formula holds only when \(a\) and \(b\) are both 0 or more. It does not work the same way for square roots of negative numbers (imaginary numbers), so do not use it when a negative number is under the radical.
Dividing radicals and rationalizing the denominator
Standard notation
(the usual math form)
\(\dfrac{\sqrt{a}}{\sqrt{b}}\)
\(=\)
\(\dfrac{\sqrt{a} \times \sqrt{b}}{\sqrt{b} \times \sqrt{b}}\)
\(=\)
\(\dfrac{\sqrt{ab}}{b}\)
In words
(symbols replaced with words)
①
a radical left in the denominator
\(=\)
②
numerator and denominator both multiplied by \(\sqrt{b}\)
\(=\)
③
the denominator is now the whole number \(b\)
The formula in words
①
For \(a\) of 0 or more and a positive \(b\) (it is a denominator, so not 0), write the division as a fraction, with
a radical left in the denominator
②
Rewrite it with the
numerator and denominator both multiplied by \(\sqrt{b}\)
The denominator becomes \(\sqrt{b} \times \sqrt{b} = b\), so the radical disappears, and
③
the denominator is now the whole number \(b\)
Quick example
Rewriting \(1 \div \sqrt{3}\) with no radical in the denominator gives
\(\dfrac{1}{\sqrt{3}}\)
\(=\)
both multiplied by \(\sqrt{3}\)
\(=\)
\(\dfrac{\sqrt{3}}{3}\)
\(\dfrac{1}{\sqrt{3}} = \dfrac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \dfrac{\sqrt{3}}{3} \approx 0.5774\)
Key idea
Multiplying the numerator and denominator by the same number does not change the value of a fraction. Rationalizing the denominator uses this to remove the radical from the denominator only. It changes the look, not the value, so \(\dfrac{1}{\sqrt{3}}\) and \(\dfrac{\sqrt{3}}{3}\) are the same number (about 0.5774).
Answers are written with no radical in the denominator because the size is easier to judge and answers are easier to compare. With \(\dfrac{1}{\sqrt{3}}\), you would have to divide 1 by 1.73…, but with \(\dfrac{\sqrt{3}}{3}\), you just divide 1.73… by 3.
The steps are the same when there are numbers in front of the radicals: divide the numbers in front, then rationalize the radical part (example: \(6\sqrt{6} \div 3\sqrt{2} = 2\sqrt{3}\)). When the denominator has two terms, such as \(\sqrt{3} + 1\), a different method is needed (multiplying by the conjugate). This calculator handles only denominators with a single radical.
Simplifying a radical means factoring the radicand into primes, finding the perfect squares, and taking their roots outside the radical (\(\sqrt{a^2b} = a\sqrt{b}\)). In addition and subtraction, only radicals with the same radicand (like terms) can be combined. In multiplication and division, the radicands can be multiplied or divided together. By convention, a quotient is written with no radical in the denominator (with the denominator rationalized).
Symbols and terms
Symbols
| \(\sqrt{\phantom{a}}\) | square root (radical sign) | The symbol for a square root, called the radical sign. It is said to come from a stylized r, the first letter of the Latin word radix, meaning "root". It stands for the positive number whose square is the number under it, as in \(\sqrt{9} = 3\). |
| \(\sqrt{a}\) | square root of a | The positive square root of \(a\). Squaring it gives back \(a\) (\(\left(\sqrt{a}\right)^2 = a\)). The number \(a\) under the radical must be 0 or more. |
| \(a\sqrt{b}\) | a root b | \(a\) times \(\sqrt{b}\) (the multiplication sign is left out). A simplified radical has this form: \(a\) is the number that came outside, and \(b\) is the number left inside. |
| \(a^2\) | a squared | \(a\) multiplied by itself (\(a \times a\)). The small 2 at the upper right shows how many times to multiply. The first job in simplifying a radical is to find this form inside it. |
| \(\approx\) | is approximately equal to | The symbol for "about equal". The value of a radical does not come out even, so when you write it as a decimal you use this symbol, as in \(\sqrt{2} \approx 1.41421356\). |
| \(p, q\) | p, q | Letters often used for the numbers in front of radicals. On this page, as in \(p\sqrt{a} + q\sqrt{a}\), they tell how many copies of \(\sqrt{a}\) there are. |
| \(<\) | is less than | The symbol meaning the left side is smaller than the right side. On this page it shows which two whole numbers a radical lies between, as in \(6 < 4\sqrt{3} < 7\). |
Terms
| square root | A number whose square is the given number. \(9\) has two square roots, \(3\) and \(-3\). The positive one is written \(\sqrt{9}\) and the negative one \(-\sqrt{9}\). |
| radical sign | The symbol √ for a square root. An expression such as \(\sqrt{5}\) is called a radical, and it stands for the positive square root of the number inside. |
| radicand | The number under the radical sign. In \(\sqrt{48}\), the radicand is 48. |
| simplest radical form | The form you get by simplifying a radical without changing its value: the perfect squares are taken outside so the radicand is the smallest possible whole number (\(\sqrt{48} \to 4\sqrt{3}\)). It also has no radical in a denominator. |
| perfect square | A number you get by squaring a whole number, such as \(1, 4, 9, 16, 25, 36, \dots\). If the radicand is a perfect square, the radical goes away and you get a whole number (\(\sqrt{36} = 6\)). |
| perfect square factor | A factor of a number that is a perfect square. \(48\) has the perfect square factor \(16 = 4^2\), so \(\sqrt{48} = 4\sqrt{3}\). Numbers whose only perfect square factor is \(1\) (\(2, 3, 5, 6, 7, 10, \dots\)) cannot be simplified further. |
| prime factorization | Writing a whole number as a product of primes only (\(48 = 2^4 \times 3\)). Each pair of a prime factor lets one copy come outside the radical, so primes with an even exponent come out completely, and primes with an odd exponent leave one copy inside. It is the sure way to find perfect squares. |
| prime number | A whole number of 2 or more that can be divided evenly only by 1 and itself, such as \(2, 3, 5, 7, 11, \dots\). Primes are the building blocks of prime factorization. |
| like terms | Terms whose variable parts are exactly the same, like \(3x\) and \(5x\). For radicals, terms with the same radicand, like \(3\sqrt{2}\) and \(5\sqrt{2}\), are like terms (also called like radicals), and only these can be combined. |
| rationalizing the denominator | Rewriting a fraction with a radical in the denominator by multiplying the numerator and denominator by the same number, so that no radical is left in the denominator. The value does not change (\(\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}\)). |
| rational number | A number that can be written as a fraction of two whole numbers. Whole numbers, terminating decimals and repeating decimals are all rational. "Rational" comes from "ratio". |
| irrational number | A number that cannot be written as a fraction. As a decimal it never ends and never repeats. \(\sqrt{2}\), \(\sqrt{3}\) and \(\pi\) are irrational. |
| real number | All the rational and irrational numbers together, which match the points on the number line. This page works with square roots within the real numbers. |
| approximation | A value close to the exact value, used in its place. In \(\sqrt{2} \approx 1.414\), the right side is an approximation. A radical never comes out even, so any decimal you write for it is an approximation. |
| imaginary number | A number that is not real. If \(i\) is the number whose square is \(-1\), the square roots of \(-4\) are \(2i\) and \(-2i\). Square roots of negative numbers are imaginary and cannot be placed on the number line. In the US they are taught with complex numbers in Algebra 2. |
| nested radical | A radical with another radical inside it, such as \(\sqrt{5 + 2\sqrt{6}}\). When the conditions are right, it can be rewritten without the nesting, as \(\sqrt{3} + \sqrt{2}\). This calculator does not handle them. |
| number line | Numbers laid out as points on a straight line. It lets you see how large a radical is and which two whole numbers it lies between. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.
| Powers and squares (Grades 6–8) |
|
| Prime factorization (Grade 6) |
|
| What a square root means (Grade 8) |
|
| Like terms in algebra (Grades 6–7) |
|
| Fractions and simplifying them (Grades 5–6) |
|
| The Pythagorean theorem (Grade 8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to check what a square root means
| Number under the radical a | 3 |
| Value of √a | =SQRT(B1) |
| √a squared (back to B1) | =B2^2 |
Table to check a simplified radical
| Number that comes outside a | 4 |
| Number left inside b | 3 |
| Original number a²×b | =B1^2*B2 |
| Value of √(a²b) | =SQRT(B3) |
| Value of a√b (the same) | =B1*SQRT(B2) |
Table to check adding radicals
| Number under the radical a | 2 |
| First coefficient p | 3 |
| Second coefficient q | 5 |
| Value of p√a + q√a | =B2*SQRT(B1)+B3*SQRT(B1) |
| Value of (p+q)√a (the same) | =(B2+B3)*SQRT(B1) |
Table to check multiplying radicals
| First number under the radical a | 3 |
| Second number under the radical b | 6 |
| Value of √a × √b | =SQRT(B1)*SQRT(B2) |
| Value of √(ab) (the same) | =SQRT(B1*B2) |
| Value of the simplified 3√2 (the same) | =3*SQRT(2) |
Table to check rationalizing the denominator
| Number under the radical b | 3 |
| Value of 1 ÷ √b | =1/SQRT(B1) |
| Value of √b ÷ b (the same) | =SQRT(B1)/B1 |
Copy a table and paste it into cell A1. The upper rows are your inputs, and the lower rows are calculated automatically. SQRT is the square root function, and "^" is a power.
The first table puts 3 in a: √3 is 1.732050808…, and squaring it gives back 3.
The second table puts 4 in a and 3 in b. The original number is 48, and √48 and 4√3 both come out as 6.928203230….
The third table is 3√2 + 5√2, and both calculations give 11.31370850…. The fourth table is √3 × √6, and all three formulas give 4.242640687….
The fifth table is 1 ÷ √3, and the rationalized √3 ÷ 3 gives the same 0.5773502692…. Change the values, and you can check that it works for any combination.
Note that Excel shows decimal approximations. Use the calculator on this page to see exact forms such as 4√3.
The first table puts 3 in a: √3 is 1.732050808…, and squaring it gives back 3.
The second table puts 4 in a and 3 in b. The original number is 48, and √48 and 4√3 both come out as 6.928203230….
The third table is 3√2 + 5√2, and both calculations give 11.31370850…. The fourth table is √3 × √6, and all three formulas give 4.242640687….
The fifth table is 1 ÷ √3, and the rationalized √3 ÷ 3 gives the same 0.5773502692…. Change the values, and you can check that it works for any combination.
Note that Excel shows decimal approximations. Use the calculator on this page to see exact forms such as 4√3.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to check what a square root means
| Number under the radical a | 3 |
| Value of √a | =SQRT(B1) |
| √a squared (back to B1) | =B2^2 |
Table to check a simplified radical
| Number that comes outside a | 4 |
| Number left inside b | 3 |
| Original number a²×b | =B1^2*B2 |
| Value of √(a²b) | =SQRT(B3) |
| Value of a√b (the same) | =B1*SQRT(B2) |
Table to check adding radicals
| Number under the radical a | 2 |
| First coefficient p | 3 |
| Second coefficient q | 5 |
| Value of p√a + q√a | =B2*SQRT(B1)+B3*SQRT(B1) |
| Value of (p+q)√a (the same) | =(B2+B3)*SQRT(B1) |
Table to check multiplying radicals
| First number under the radical a | 3 |
| Second number under the radical b | 6 |
| Value of √a × √b | =SQRT(B1)*SQRT(B2) |
| Value of √(ab) (the same) | =SQRT(B1*B2) |
| Value of the simplified 3√2 (the same) | =3*SQRT(2) |
Table to check rationalizing the denominator
| Number under the radical b | 3 |
| Value of 1 ÷ √b | =1/SQRT(B1) |
| Value of √b ÷ b (the same) | =SQRT(B1)/B1 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the numbers with your own.
How to calculate it in Python
from math import isqrt
def simplify_radical(n):
# Rewrite √n as a√b (b has no perfect square factor) and return (a, b)
outside = 1
inside = n
d = 2
while d * d <= inside:
while inside % (d * d) == 0:
inside //= d * d
outside *= d
d += 1
return outside, inside
outside, inside = simplify_radical(48)
if inside == 1:
print(f"√48 = {outside}")
else:
print(f"√48 = {outside}√{inside}")
# Check with decimal values too (they are the same)
print(48 ** 0.5, outside * inside ** 0.5)
# If the radicand is a perfect square, the radical goes away
print(simplify_radical(36), isqrt(36))
It uses only the standard library. Try d = 2, 3, 4, … in turn. While the number can be divided by d squared, keep dividing, and move one d outside the radical each time. Running it prints "√48 = 4√3", then 6.928203230275509 twice, which shows the value is the same before and after simplifying. The last line is √36, where the radical goes away, and it prints (6, 1) and 6. Change the numbers and try it.
How to write it in LaTeX and other math languages (copy and paste)
What a square root and the radical sign mean
(√a)² = a
\left(\sqrt{a}\right)^{2} = a
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup>
<mrow><mo>(</mo><msqrt><mi>a</mi></msqrt><mo>)</mo></mrow>
<mn>2</mn>
</msup>
<mo>=</mo>
<mi>a</mi>
</mrow>
</math>
(sqrt(a))^2 = a
Sqrt[a]^2 == a
sqrt(a)^2 = a;
sqrt(a)^2 == a
(√a)^2 = a
Simplifying a radical (take the perfect square out)
√(a²b) = a√b
\sqrt{a^{2}b} = a\sqrt{b}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi></mrow></msqrt>
<mo>=</mo>
<mi>a</mi>
<msqrt><mi>b</mi></msqrt>
</mrow>
</math>
sqrt(a^2 b) = a sqrt(b)
Sqrt[a^2 b] == a Sqrt[b]
sqrt(a^2*b) = a*sqrt(b);
sqrt(a^2*b) == a*sqrt(b)
√(a^2 b) = a√b
Adding and subtracting radicals (combine like terms)
p√a + q√a = (p + q)√a
p\sqrt{a} + q\sqrt{a} = \left(p + q\right)\sqrt{a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>p</mi><msqrt><mi>a</mi></msqrt>
<mo>+</mo>
<mi>q</mi><msqrt><mi>a</mi></msqrt>
<mo>=</mo>
<mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo>
<msqrt><mi>a</mi></msqrt>
</mrow>
</math>
p sqrt(a) + q sqrt(a) = (p + q) sqrt(a)
p Sqrt[a] + q Sqrt[a] == (p + q) Sqrt[a]
p*sqrt(a) + q*sqrt(a) = (p + q)*sqrt(a);
p*sqrt(a) + q*sqrt(a) == (p + q)*sqrt(a)
p√a + q√a = (p + q)√a
Multiplying radicals (multiply the insides to make one)
√a × √b = √(ab)
\sqrt{a} \times \sqrt{b} = \sqrt{ab}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msqrt><mi>a</mi></msqrt>
<mo>×</mo>
<msqrt><mi>b</mi></msqrt>
<mo>=</mo>
<msqrt><mrow><mi>a</mi><mi>b</mi></mrow></msqrt>
</mrow>
</math>
sqrt(a) * sqrt(b) = sqrt(a b)
Sqrt[a] Sqrt[b] == Sqrt[a b]
sqrt(a)*sqrt(b) = sqrt(a*b);
sqrt(a)*sqrt(b) == sqrt(a*b)
√a × √b = √(ab)
Dividing radicals and rationalizing the denominator
√a ÷ √b = √(ab) / b
\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a}\sqrt{b}}{\sqrt{b}\sqrt{b}} = \frac{\sqrt{ab}}{b}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfrac><msqrt><mi>a</mi></msqrt><msqrt><mi>b</mi></msqrt></mfrac>
<mo>=</mo>
<mfrac>
<mrow><msqrt><mi>a</mi></msqrt><msqrt><mi>b</mi></msqrt></mrow>
<mrow><msqrt><mi>b</mi></msqrt><msqrt><mi>b</mi></msqrt></mrow>
</mfrac>
<mo>=</mo>
<mfrac><msqrt><mrow><mi>a</mi><mi>b</mi></mrow></msqrt><mi>b</mi></mfrac>
</mrow>
</math>
sqrt(a)/sqrt(b) = (sqrt(a) sqrt(b))/(sqrt(b) sqrt(b)) = sqrt(a b)/b
Sqrt[a]/Sqrt[b] == Sqrt[a b]/b
sqrt(a)/sqrt(b) = sqrt(a*b)/b;
sqrt(a)/sqrt(b) == sqrt(a*b)/b
√a/√b = √(ab)/b
How to have ChatGPT do the calculation
You are a math calculation assistant for square roots. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). 1. Simplify √48 so that the number under the radical is the smallest possible whole number. 2. Also show the prime factorization of 48 used along the way. 3. Give 3√2 + 5√2, 2√6 × 3√2 and 6 ÷ √3 each in simplest radical form (for division, with no radical in the denominator). In Python, write your own function that factors the radicand into primes and takes out the perfect squares, and check that the decimal values before and after simplifying match. Show the code you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
DataChef Features
Easy and Free
Unlimited conversions for free.
No technical knowledge required.
Intuitive and user-friendly operation.
No technical knowledge required.
Intuitive and user-friendly operation.
No Registration Required
Available immediately after access.
Can be used without registering personal information.
Can be used without registering personal information.
Safe and Secure
Fully SSL encrypted communication.
Automatic file deletion by clicking "download".
Automatic file deletion by clicking "download".
Fast
High-speed site access
and rapid file conversion.
and rapid file conversion.
No Watermark
No watermark.
No attribution required.
No attribution required.
Commercial Use Available
Free for commercial use.
No need to contact us for commercial use permission.
No need to contact us for commercial use permission.
Related Features
Nth Root Calculator (Square Root, Cube Root, nth Root)
Find square roots such as √27, cube roots and nth roots as decimals. Odd roots of negative numbers work too
Prime Factorization Calculator
Break a whole number into prime factors. It is the basis for finding the perfect squares used to simplify radicals
Quadratic Formula Calculator (Discriminant and Complex Roots)
Enter the coefficients a, b and c to solve a quadratic equation. Radicals in the answer are shown in simplest form
Ad 
