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Square roots are the most common type of radical used. A square root “unsquares” a number. For example, because we say the square root of 25 is 5. The square root of 25 is written as .
EXAMPLE
Square Root | Square Powers |
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Not possible |
Not all numbers have a nice even square root. For example, if we found on our calculator, the answer would be 2.828427124746190097603377448419... and even this number is a rounded approximation of the square root. Decimal approximations will work in most cases, but you may need the exact value, in which case you will express using the radical symbol, rather than expressing as a decimal.
Just like square roots undo squaring a number, cube roots undo cubing a number. For example, because , we say that the cube root of 8 is 2. The cube root of 8 is written as with a 3 as the index of the radical, to indicate a cube root.
EXAMPLE
Notice that the cube root of a negative number is results in a real number. This is because a negative number cubed is a negative number. So while square roots of negative numbers are non-real numbers, cube roots of negative numbers are real numbers. This actually holds true for all even roots and odd roots.
While square and cube roots are the most common type of radical we work with, we can take higher roots of numbers as well: cube roots, fourth roots, fifth roots, etc. Consider this definition of radicals:
The small letter m inside the radical is called the index. It tells us which root we are taking, or which power we are “un-doing”. For square roots the index is 2. As this is the most common root, the two is not usually written.
Take a look at several higher roots:
EXAMPLE
Higher Roots | Higher Powers |
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Not possible |
This last example is not possible, or undefined, because if we take any positive or negative number to the fourth power, the answer will be positive (or zero). or . Thus we can only take the fourth roots of non-negative numbers.
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License