Fundamentals of Mathematics, 2008a

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3.2.4 Roots
CHAPTER 3.
EXPONENTS, ROOTS, AND FACTORIZATION OF WHOLE
NUMBERS
In the English language, the word "root" can mean a source of something. In mathematical terms, the word
"root" is used to indicate that one number is the source of another number through repeated multiplication.
Square Root
We know that 49 = 7 2 , that is, 49 = 7 · 7. Through repeated multiplication, 7 is the source of 49. Thus, 7 is
a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square
root of 49.
Cube Root
We know that 8 = 2 3 , that is, 8 = 2 · 2 · 2. Through repeated multiplication, 2 is the source of 8. Thus, 2 is a
root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.
We can continue this way to see such roots as fourth roots, fth roots, sixth roots, and so on.
3.2.5 Reading Root Notation
There is a symbol used to indicate roots of a number. It is called the radical sign n √
The Radical Sign n √
The symbol n √ is called a radical sign and indicates the nth root of a number.
We discuss particular roots using the radical sign as follows:
Square Root
2√
number indicates the square root of the number under the radical sign. It is customary to drop the 2 in
the radical sign when discussing square roots. The symbol √ is understood to be the square root radical
sign.
√
49 = 7 since 7 · 7 = 7 2 = 49
Cube Root
3√
number indicates the cube root of the number under the radical sign.
3√
8 = 2 since 2 · 2 · 2 = 2 3 = 8
Fourth Root
4√
number indicates the fourth root of the number under the radical sign.
4√
81 = 3 since 3 · 3 · 3 · 3 = 3 4 = 81
In an expression such as 5√ 32
Radical Sign
√
is called the radical sign.
Index
5 is called the index. (The index describes the indicated root.)
Radicand
32 is called the radicand.
Radical
5√
32 is called a radical (or radical expression).
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