Radical Sign
The radical sign may be one of the stranger-looking mathematical symbols, but its strange looks should not fool anyone. The radical sign has a straightforward meaning.
The sign that is used when finding the square root or cube root, , is called the radical sign. A radical sign with no index number written in the notch indicates a square root. A square root can also be written with an index of , but usually the 2 is understood rather than expressly written. A radical sign with an index of 3 is written as , which indicates a cube root.
For example, to calculate the area of a square (in which all sides are equal), multiply the length of one side s by itself (squaring), so the area is s 2. If the area of a square is known but not the length of the side, the inverse of squaring will find the length of the side. The inverse operation of squaring is finding the square root. For example, if the area of a square is 16 ft2 (square feet), finding the square root is in effect asking "what number multiplied by itself, or squared, gives a result of 16?" The square root of 16 ft2 is or 4 ft, since 42 = 16.
Similarly, suppose a cube-shaped box has a volume of 125 cm3 (cubic centimeters). How can one find the length of a side, s, without measuring the box? To find the volume of a cube, s is multiplied by itself three times or raised to the third power, also called cubing the side. So the volume of a cube with side length s is s 3. The inverse operation of cubing is finding the cube root. Because 53 = 125, the cube root of 125 cm3 is or 5 cm.
One can also find the fourth root, fifth root, and so forth, by indicating which root by changing the index. So means to find the fourth root, and means to find the fifth root. Just as taking the square root is the inverse operation of squaring, and taking the cube root is the inverse operation of cubing, finding is the inverse of raising x to the fourth power, and so on.
Look at the expression (the seventh root of x ) as an example. In this expression, the integer in front of the radical sign, 7, is known as the index. The number under the radical sign, in this case x, is called the radicand .
It is helpful to remember that is usually called "square" root instead of "second" root, and is called the "cube" root instead of "third" root. If the index of a radical is greater than 3, one simply says fourth root, fifth root, and so on.
When solving the equation x 2 = 4, there are two solutions: 2 and −2. Does the expression also have two solutions: 2 and −2? No. Although (2)2 = 4 and (−2)2 = 4, the mathematical expressions = x and x 2 = 4 are not equivalent; that is, these two equations do not have the same solution set. The solution set for = x is 2 and the solution set for x 2 = 4 is 2 and −2.
To verify this, use a calculator to find the square root of 4. On most calculators, the answer you get will be simply 2. Is the calculator wrong? The short answer to this question is no. When using the radical sign, the expression is understood to mean positive square root of 4. When both solutions to a square root are wanted, the radical must have the symbol ± in front of it. So, , and and −2.
The answer to a problem when finding a square root also depends on the context of the problem. When using a square root to find the side length of a square table for which only the area is known, a negative value does not make sense. A table with an area of 16 ft2 would not have a side length of 4 feet! When solving problems using the radical sign, write the radical sign alone if only the positive root is desired, and write ± in front of a radical sign when both roots are desired.
When working within the real number system (or the numbers that can be found on a real number line), finding roots that have an even number for the index—such as square roots, fourth roots, sixth roots, and so forth—the radicand must be greater than or equal to 0 in order to get an answer that is part of the real number system.
For example, consider the expression . To solve this, one needs to find out which number, when multiplied by itself, equals −4. Consider this: 2 × 2 = 4, and −2 × (−2) = 4. Clearly there is no number within the real number system that when multiplied by itself equals −4. Therefore, when simplifying a radical that has an even index and a radicand that is less than 0, the answer is undefined. However, this does not mean that it is impossible to find the square root of −4. This root does exist, but it is not defined within the real number system.
What about Is this solution defined in the real number system? To simplify this radical, a number which when multiplied by itself three times equals 27 must be found. It is known that 33 = 27 because 3 × 3 × 3 = 27. What about (−3)3? Because (−3) × (−3) × (−3) = −27, the cube root of −27 is −3. Therefore, it is possible to find the root of a radicand that is less than 0 when the index is an odd number (three, five, seven, and so on).
Radicals can also be used to express irrational numbers . An irrational number is one that cannot be expressed as a ratio of two integers, such as 1/2. An example of an irrational number is . In decimal form, expands into 1.414213562…. Because this number cannot be written as afraction, and because the decimal continues without repeating, it is often better to use the radical to express the number exactly as
see also Numbers, Complex; Numbers, Irrational; Powers and Exponents.
Max Brandenberger
Bibliography
Gardner, Martin. The Night is Large: Collected Essays 1938–1995. New York: St. Martin's Griffin, 1996.
Paulos, John Allen. Beyond Numeracy: Ruminations of a Numbers Man. New York: Alfred A. Knopf, 1991.
———. Innumeracy: Mathematical Illiteracy and its Consequences. New York: Hill and Wang, 1998.
