No one knows for sure where the square root symbol “√” first appeared. The earliest sighting of it used was in Babylon in 1600 BC, but it also appeared in Greece, China, India, and Egypt.
Scholars today think square roots come from ‘r,’ the first letter of Latin word Radix. Radix means root. Other people believe that it comes from the Arabian letter ج which also appears in root.
Initially, the √ symbol didn’t have the ‘-‘ bar that usually sits over numbers within the symbol. That line is Latin for bond and appears often in written form, but not on the internet. Notations of higher degrees such as ∛ are supposedly from mathematician Albert Girard.
You might wonder why the square root is called root, but an equation can explain that. Let’s say you write x = ⁿ√a as xⁿ = a. You would call x the root (radical) because it’s the hidden base of a.
The most common things you do with numbers in mathematics is add, subtract, divide, and multiply. You then start introducing trigonometry, exponentiation, and square roots.
Explaining the definition of a square root is straightforward. The square root of a number (x) is a number (y) that yields x once you square it. It looks like this as a formula: y² = y x y.
You can then express the square root formula as:
√x = y ⟺ x = y²
The arrow symbol ⟺ has come to mean ‘if, and only if.’
Every positive number has two square roots – one positive, one negative. To streamline the process, we would only ever use a positive one. Zero doesn’t have two square roots because of √0 = 0 which is not positive or negative.
When you start to find yourself entwined in complicated equations, another notation of square roots applies. A second square root formula exists where a square root of a number is raised to the exponent of a fraction by one half. The square root formula looks like this:
√x = x^ (1/2) = x^ (0.5)
Using square roots geometrically offers a different definition. The square root of a square’s area offers its side measurement. Have you ever wondered why square root has ‘square’ in its name? That’s why.
The same rules apply with a cube root (∛). If you cube root a cube’s volume, you end up with the edge’s length. You use a square root for surface areas, but you use a cube root for volume and density.
In most cases, you can access a square root calculator to find the square root of a number. If you only have yourself and your brain, it pays to know some basic square root sums.
Keep this information locked in your brain:
Square root of 1: √1 = 1
(1 x 1 = 1)
Square root of 4: √4 = 2
(2 x 2 = 4)
Square root of 9: √9 = 3
(3 x 3 = 9)
Square root of 16: √16 = 4
(4 x 4 = 16)
Square root of 25: √25 = 5
(5 x 5 = 25)
Square root of 36: √36 = 6
(6 x 6 = 36)
Square root of 49: √49 = 7
(7 x 7 = 49)
Square root of 64: √64 = 8
(8 x 8 = 64)
Square root of 81: √81 = 9
(9 x 9 = 81)
Square root of 100: √100 = 10
(10 x 10 = 100)
Square root of 121: √121 = 11
(11 x 11 = 121)
Square root of 144: √144 = 12
(12 x 12 = 144)
These numbers above are some of the simplest for remembering the square root. However, there are going to be cases when it’s not so easy. There are many ways to work out the square root of a number without your memory or a square root calculator.
For example, let’s say you want to work out the square root of 32.
1. You know that, for natural numbers such as 25 that √25 = 5, and √36 = 6, so your answer for √32 will be between the numbers 5 and 6. Take a guess, as it’s more likely to be toward 6. Say that √32 = 5.5.
2. Square 5.5 (√5.5) to get 30.25, which is lower than 32. Aim for a higher number.
3. Square 5.6 (√5.6) to get 31.36. The number is higher and closer to 32.
4. You can carry on getting as close as possible, or you can be happy with the accuracy of that figure as it is.
If this method doesn’t work out, there is another. Simplify the square root first. Then, use approximate figures of the prime numbers’ square roots. These are generally rounded to two decimals:
Square root of 2: √2 = 1.41
Square root of 3: √3 = 1.73
Square root of 5: √5 = 2.24
Square root of 7: √7 = 2.65
Square root of 11: √11 = 3.32
Square root of 13: √13 = 3.61
Square root of 17: √17 = 4.12
Square root of 19: √19 = 4.34
If that doesn’t make sense, then consider this example when looking for the square root of 52 (7.22). Simplify √52 to 2√13. Substitute √13 = 3.61. The next step is to multiply it in the formula:
√52 = 2 x 3.61 = 7.22
Another helpful hint is to use a prime number calculator to know if a number is prime. A prime number is higher than one. It’s also a natural number that exists without being the product of two other natural numbers. The number 7 is prime as you can multiply 7 x 1 or 1 x 7. The number 8 is not, as aside from multiplying it by itself, you can use 2 x 4 and 4 x 2.
You are not always going to need to know a square root’s exact answer. In such a case, a square root calculator is all you need to get a quick estimate. The estimates are still close to the real answer. For example, you want to know if 3√4 is greater than 7. You know that √4 = 2. In that case, 3√4 = 3 x 2 = 6. It’s not greater than 7.
You may also find it convenient to know that a square root calculator offers answers for both fields. You can fill in the number to get the square root, or square root to get the number. When you move out of the realm of mathematics and into science, the format starts to change.
Instead of displaying answers like before, you present them as scientific notations. The answer would have a point between any non-zero numbers with a decimal multiplied by ten that is raised to an exponent.
Take, 0.00498, for example. In scientific notation, it’s 4.98 x 10⁻3. You can convert results from a square root calculator using a scientific notation calculator.
Learning how to simplify square roots only works if you choose a number that forms a perfect square. For example, 4, 9, 16 and 26 are perfect squares. The reason for this is that you can express them as 2², 3², 4², and so on.
If you’re struggling to understand the concept, then here are a couple of examples.
Can you simplify 27? Use a factor calculator to work out 27’s factors. They are 1, 3, 9, and 27. The existence of nine means you can.
What about 15? The number 15 has factors of 1, 3, 5, and 15. You can’t form a perfect square, so you can’t simplify it.
It can be a challenge to know if you can simplify square roots if you don’t know how it works. The alternative square root formula comes into play here:
√x = x^ (1/2)
You can switch between both square root forms whenever you need. It’s also important to know that two numbers’ power of multiplication is the same as those numbers raised to the powers. The formula looks like this:
(x * y) ^ (1/2) = x^ (1/2) x y^ (1/2) ⟺ √ (x x y) = √x x √y,
Calculating a square root to find its perfect square is not going to be effortless every time. If you dig deep, you may find a perfect square hidden in the factors of a number. You can then write it as a multiplication of two numbers. One number is a perfect square, such as 45 = 9 x 5. Nine is the perfect square.
The rule of using this formula is that at least one factor should be a perfect square. Otherwise, you can’t simplify the square root. From here, you need to add the multiplication to the square root.
√45 = 45^ (1/2) = (9 x 5) ^ (1/2) = 9^ (1/2) x 5^(1/2) = √9 x √5 = 3√5
It might seem like a redundant process to simplify a square root, but now you understand it! Now, all you need to know is that a square root is the same as the power of one half. There are some more fun examples below.
How do you simplify the square root of 27?
√27 = √ (9 x 3) = √9 x √3 = 3√3
What about simplifying the square root of 8?
√8 = √ (4 x 2) = √4 x √2 = 2√2
Now into challenging territory: what is the square root simplified of 144?
144 = √ (4 x 36) = √4 x √36 = 2 x 6 = 12
As you see from that example, 144 did not need simplifying. It’s a perfect square! All you need to know in that case is that 12 x 12 = 144. The goal of that example was to establish that even large numbers can be perfect squares.
What about cube roots or simplifying roots of higher orders? With cube roots, you need to find a factor that forms a perfect cube, such as 8 = 2³ or 27 = 3³. Once you see that factor, split the number in half under the cube root.
An example is when you simplify ³√192.
∛192 = ∛ (64 x 3) = ∛64 * ∛3 = 4∛3
Don’t let the complex looking equation frazzle you! It takes practice, but the simplification process is straightforward.
The following information covers adding, subtracting, multiplying and dividing square roots.
Addition and subtraction
You will discover early on that square root addition and subtraction is more involved than with regular numbers. Everyone knows that 1 + 2 = 3, but √1 + √2 doesn’t equal √3. It might not make sense why that is right away, but they are two different equations.
It would be like trying to add a triangle to a square. Nothing happens when you add them together. You still have a triangle and a square. If you had one square then added five more though, you would then have six squares.
The same rules apply with square roots. √1 + √2 equals √1 + √2 with no further simplification. If those square roots have the same number under its root symbol, you can add those together. An example of that process would be:
1√2 + 3√2 = 4√2
Other examples to help are:
4√11 + 6√11 = 10√11
2√2 + 3√8 = 2√2 + 3√8 = 2√2 + 6√2 = √8√2.
In this equation, you were able to simplify the √8 = √(4 x 2) = √4 x √2 = 2√2
Multiplication and Division
Multiplication and division follow a similar formula to above.
√x = x^ (1/2)
You then multiply numbers that multiply to the same power:
xⁿ * yⁿ = (x x y)ⁿ
Which means:
x^(1/2) x y^(1/2) = (x x y)^(1/2) ⟺ √x x √y = √(x x y)
Unlike addition, you can multiply every two square roots. The order of those numbers also doesn’t matter. Organize the formulas in your head with the following examples.
The answer for √3 x √2 = √6
The answer for 2√5 x 5√3 = 2√5 x 5√3 = 2 x 5 x √5 x √3 = 10√15
The square root division process is nearly the same.
x^(1/2) / y^(1/2) = (x / y)^(1/2) ⟺ √x / √y = √(x / y)
There are two differences. The first is that you replace x with /. Division is also not a commutative operator. Numbers under the square root and standing numbers are separate. Examples of this below are:
√15 / √3 = √15 / √3 = √5
10√6 / 5√2 = 10√6 / 5√2 = (10 / 5) x (√6 / √2) = 2√3
You can use your previous knowledge to understand using exponents and fractions with square roots. The first thing you need to know is that you will use the alternative square root form.
This is:
√x = x^ (1/2)
You should also understand the power rule of:
(x^n) ^ m = x^(n*m)
N and M = real numbers
If you use ½ instead of M, you will only get a square root:
√(x^n) = (x^n) ^ (1/2) = x^(n/2)
If that seems a little hard to understand, consider these examples:
What is the square root of 2^4?
√(2^4) = (2^4)^(1/2) = 2^(4/2) = 2^2 = 4
What is the square root of 5^3?
√(5^3) = (5^3)^(1/2) = 5^(3/2)
Now that you understand exponents, what can we say about fractions? Fractions closely relate to the division section of square roots above.
(x / y)^(1/2) ⟺ √x / √y = √(x / y)
x/y = fraction
Examples of square roots in a fraction situation are below:
What is the square root of 4/9?
√ (4/9) = √4 / √9 = 2/3
What is the square root of 1/100?
√ (1/100) = √1 / √100 = 1/10
You use functions a lot in math, but you can also use them in finance, physics, and statistics. Function, appearing as f(x) on your calculator, is a formula to establish a value change of f(x) with x.
The square root function is f(x) = √x. It’s continuous and growing for a non-negative x, differentiable for a positive x, and nears the limit of infinity with (lim √x → ∞ when x → ∞). The square root function is also a real number for non-negative x and a complex one for negative x.
A function’s derivative establishes the speed of a function that changes with an argument. Take, for example, physics and an object’s position and velocity. If the function is x (t), you know it represents a car’s speed and the time with it. What determines the increase? The car’s speed. The derivative of the position is the velocity.
Derivatives normally look like this:
v (t) = x’ (t), or v (t) = dx (t) / dt
The calculation process can be challenging with the general function’s derivative. Formulas for it are:
f(x) = x^n
n = real number
f'(x) = n x x^(n-1)
That’s the derivative of a square root.
The alternative form of a square root is:
√x = x^ (1/2)
When n = ½, the derivative of a square root is:
(√x)' = (x^(1/2))' = 1/2 x x^(-1/2) = 1/(2√x).
The derivation will involve fractions because the number to the negative power is one over the number.
If you want to know the coefficients of the Taylor expansion, you need to know the derivative of the square root. The Taylor Series lets you gain approximations for functions with the much-easier-to-calculate polynomials.
An example is below.
With the point x = 0 and a Taylor expansion of √(1 + x):
√(1 + x) = 1 + 1/2 x x - 1/8 x x² + 1/16 x x³ - 5/128 x x⁴ + __
The equation is valid for -1 ≤ x ≤ 1, even though the expression above has endless terms. If you tried it with x = 0.5, it would be:
√(1.5) = 1 + 1/2 x0.5 - 1/8 * 0.25 + 1/16 x 0.125 - 5/128 x 0.0625,
And approximation of: √ (1.5) = 1.2241
The real answer is √(1.5) ≈ 1.2247, so the estimation is not far off.
Are you thrilled or riveted? Read on to discover how to calculate the square root of a negative number.
With regular numbers, there is no square root of negative numbers. Your school teachers would have taught you as much as well. When mathematicians wanted to perform advanced calculations, they needed more numbers. Thus , they introduced complex numbers like this:
x = a + b x i
x = complex number
a = complex number’s real part
b = complex number’s imaginary part
i = the difference between the real and imaginary number
Complex numbers can look like this:
2 + 3i, 5i, 1.5 + 4i, 2
2 is a real number, but it’s a complex number when b = 0. You can define complex numbers as a real number generalization.
‘i’ is an imaginary number, but it has meaning. As such, it’s valid in this equation:
i = √ (-1)
Finally, some examples so you can put all that new knowledge to the test.
What is the square root of -9?
√(-9) = √(-1 x 9) = √(-1)√9 = 3i
What is the square root of -49?
√(-49) = √(-1 x 49) = √(-1)√49 = 7i
It’s not as hard as it looks, but you also can’t use this formula with a cube root. Cube root doesn’t run into the same problems as square root because you can obtain a negative number with three identical numbers.
