Finding Square Roots, Cube Roots, and nth Roots in Excel

Using Exponents and the SQRT Function to Find Square and Cube Roots in Excel

Finding Square Roots, Cube Roots, and nth Roots in Excel Using Exponents
© Ted French

In Excel, square roots of numbers can be found by:

    • Creating a formula using exponents (rows two and three above)
    • Using the SQRT function (rows five and six above)
  • Using a formula containing exponents or powers to determine cube roots

The SQRT Function's Syntax and Arguments

A function's syntax refers to the layout of the function and includes the function's name, brackets, comma separators, and arguments.

The syntax for the SQRT function is:

= SQRT (Number)

Number (required)-The number for which square root must be found:

  • Can be any positive number or a cell reference to the location of the data in a worksheet
  • If a negative value is entered for the Number argument, SQRT returns the #NUM! error value (row 7 in the image above)

Since multiplying two positive or two negative numbers together always returns a positive result, it is not possible to find the square root of a negative number such as (-25) in the set of real numbers.

SQRT Function Examples

In rows 5 through 8 in the image above, various ways of using the SQRT function in a worksheet are shown.

The examples in rows 5 and 6 show how the actual data can be entered as the Number argument (row 5) or the cell reference for the data can be entered instead (row 6).

The example in row 7 shows what happens if negative values are entered for the Number argument, while the formula in row 8 uses the ABS (absolute) function to correct this problem by taking the absolute value of the number before finding the square root. 

The order of operations requires that Excel always perform calculations on the innermost pair of parentheses first and then work its way out, thus the ABS function must be placed inside SQRT for this formula to work.

Entering the SQRT Function

Options for entering the SQRT function include manually typing in the entire function: 

=SQRT(A6) or =SQRT(25)

or using the function's dialog box - as outlined below:

  1. Click on cell C6 in the worksheet - to make it the active cell;
  2. Click on the Formulas tab of the ribbon menu;
  3. Choose Math & Trig from the ribbon to open the function drop down list;
  4. Click on SQRT in the list to bring up the function's dialog box;
  5. In the dialog box, click on the Number line;
  6. Click on cell A6 in the spreadsheet to enter this cell reference as the Number line argument;
  7. Click OK to close the dialog box an return to the worksheet;
  8. The answer 5 (the square root of 25) should appear in cell C6;
  9. When you click on cell C6 the complete function = SQRT ( A6 ) appears in the formula bar above the worksheet.

Exponents in Excel Formulas

The exponent character in Excel is the caret (^) located above the number 6 on standard keyboards.

Exponents - such as 52 or 53 - therefore, are written as 5^2 or 5^3 in Excel formulas.

To find square or cube roots using exponents, the exponent is written as a fraction or decimal (rows two, three, and four in the image above).

The formulas =25^(1/2) and =25^0.5 find the square root of 25 while =125^(1/3) finds the cube root of 125. The result for all formulas is 5 (cells C2 to C4 in the example).

Finding nth Roots in Excel

Exponent formulas are not restricted to finding square and cube roots. The nth root of any value can be found by entering the desired root as a fraction after the carat character in the formula.

In general, the formula looks like this:

=value ^(1/n)

where value is the number you want to find the root of and n is the root. So,

  • the fourth root of 625 would be written: 625^(1/4);
  • the tenth root of 9,765,625 would be written: 9765625^(1/10).

Bracketing Fractional Exponents

Notice, in the formula examples above, that when fractions are used as exponents they are always surrounded by parentheses or brackets.

This is done because the order of operations Excel follows in solving equations carries out exponent operations before division - the forward slash ( / ) being the division operator in Excel.

So if the parenthesis is left out, the result for the formula in cell B2 would be 12.5 rather than 5 because Excel would:

  1. raise 25 to the power of 1
  2. divide the result of the first operation by 2.

Since any number raised to the power of 1 is just the number itself, in step 2, Excel would end up dividing the number 25 by 2 with the result being 12.5.

Using Decimals in Exponents

One way around the above problem of bracketing fractional exponents is to enter the fraction as a decimal number, as shown in row 3 in the image above.

Using decimal numbers in exponents works well for certain fractions where the decimal form of the fraction doesn't have too many decimal places - such as 1/2 or 1/4 which in decimal form are 0.5 and 0.25 respectively.

The fraction 1/3, on the other hand, which is used to find the cube root in row 3 of the example, when written in decimal form gives the repeating value: 0.3333333333...

To get an answer of 5 when finding the cube root of 125 using a decimal value for the exponent would require a formula such as: