How to use the SQRT function
What is the SQRT function?
The SQRT function calculates the positive square root.
The SQRT function is not the only method to calculate the square root in an Excel formula. The caret symbol lets you raise a number to a given power, using 0.5 as a power is the same as the square root.
For example, SQRT(100) equals 10, however, 100^0.5 also equals 10.
The SQRT function can only calculate the square root, no other nth roots.
Table of Contents
1. Introduction
What is a square root?
The square root of x is denoted by √x but it can also be x1/2 or x0.5 A square root of a number x is a value that when multiplied by itself equals x.
√x * √x = x
For example
√4 * √4 = 4
It is the reverse operation of squaring a number.
x2 = 4
x = √4
x1 = +2
x2 = -2
Why are there two solutions to a square root?
There are two solutions to a square root is because when you square a number whether it is positive or negative you get the same result.
For example, (-2)2 = 22 = 4.
If you want to find the number that was squared to get 4, you have two possible answers:
√4 = 2 or √4 = -2.
Often we only want the positive solution of a square root, like when we are dealing with length, area, or time.
Is there a solution to a square root of a negative number?
No, there is no real number solution for the square root of a negative number. For example: √(-16) has no solution because if you square a real number you always get a positive result:
(-4)2 = 16
4^2 = 16
So no real number squared can produce a negative result.
However, we can find a complex number solution by introducing the imaginary unit i:
i = √(-1)
Where i is defined such that i2 = -1. Using i we can take square roots of negatives:
√(-16) = 4i
Because (4i)(4i) = 16*i2 = 16*-1 = -16
Square roots of negative real numbers are undefined, the concept of imaginary numbers and the imaginary unit i allows extending square roots to negative and complex numbers.
Check out the Engineering category for Excel functions dealing with complex numbers involving imaginary units.
What is an even power?
An even power or exponent refers to when a number is raised to an even integer exponent. Even powers have the general form x2n where n is some integer.
The first even power is the square x2. Higher even powers are x2, x4, x6, etc.
What is an even number?
An even number is an integer (whole number) that is divisible by 2 with no remainder.
2. SQRT Function Syntax
SQRT(number)
| number | Required. The positive numerical value for which you want the square root. |
3. Example
The first example in cell B3 and C3 demonstrates what you get if you try to square root a negative number in Excel. The formula returns the #NUM! error value if a number is negative.
Formula in cell C3:
Use the ABS function to remove the sign if a number is negative.
The second example in cell B4 and C4 calculates the square root of 0 (zero) which is 0 (zero).
The third example in cell B5:C5 calculates the square root of 1 which is equal to 1.
The fourth example in cell B6:C6 calculates the square root of 4 which is equal to 2. 2*2 = 4
The fifth example in cell B7:C7 calculates the square root of 9 which is equal to 3. 3 * 3 = 9
The sixth example in cell B8:C8 calculates the square root of 36 which is equal to 6. 6 * 6 = 36
4. Calculate the hypothenuse in a right triangle
The SQRT function lets you calculate the hypotenuse using the Pythagorean Theorem:
c = √(a2 + b2)
Excel formula in cell C5:
The formula in cell C5 calculates the hypotenuse in a right triangle based one side equal to 6 units and the other side equal to 8 units. A right triangle must have one angle equal to 90 degrees.
What is the Pythagorean Theorem?
The Pythagorean Theorem is a mathematical relationship that describes the lengths of the sides of a right triangle.
In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
a2 + b2 = c2
a and b are the lengths of the legs of the triangle.
c is the length of the hypotenuse.
To calculate the hypothenuse the formula becomes:
c = √(a2 + b2)
Explaining formula
Step 1 - Square leg 1
The caret character lets you raise a number to a given power. The caret symbol is only one character compared to the POWER function that also raises a number to a given power.
C2^2
becomes
6^2
and returns 36 (6 * 6 = 36).
Step 2 - Square leg 2
C3^2
becomes
8^2
and returns 64 (8 * 8 = 64).
Step 3 - Add the squared numbers
C2^2+C3^2
becomes
36 + 64 equals 100
Step 4 - Calculate the square root of the sum
SQRT(C2^2+C3^2)
becomes
SQRT(100)
and returns 10.
5. Calculate the legs in a right triangle
The SQRT function lets you calculate the legs of a right triangle using the Pythagorean Theorem:
a = √(c2 - b2)
or
b = √(c2 - a2)
Excel formula in cell C5:
The formula in C5 calculates the length of leg 1 based on leg 2 equal to 8 units and the hypotenuse equal to 10.
What is the Pythagorean Theorem?
The Pythagorean Theorem is a mathematical relationship that describes the lengths of the sides of a right triangle.
In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
a2 + b2 = c2
a and b are the lengths of the legs of the triangle.
c is the length of the hypotenuse.
To calculate the leg of a right triangle the formula becomes:
a = √(c2 - b2)
or
b = √(c2 - a2)
Explaining formula
Step 1 - Square the hypotenuse
The caret character lets you raise a number to a given power. The caret symbol is only one character compared to the POWER function that also raises a number to a given power.
C2^2
becomes
10^2
and returns 100 (10 * 10 = 100).
Step 2 - Square leg 2
C3^2
becomes
8^2
and returns 64 (8 * 8 = 64).
Step 3 - Subtract the squared hypotenuse with the squared leg 2
C2^2-C3^2
becomes
100 - 63 equals 36
Step 4 - Calculate the square root of the sum
SQRT(C2^2+C3^2)
becomes
SQRT(36)
and returns 6.
Functions in 'Math and trigonometry' category
The SQRT function function is one of 62 functions in the 'Math and trigonometry' category.
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