Radical

Pronunciation: /ˈræd ɪ kəl/ ?

A radical is a root of a number or expression, such as square root of 12. Definition: a=b radical(c) if and only if a^b = c. The word radical can also be used to indicate the symbol indicating root, such as square root symbol.. When used as an adjective, the word radical means 'having to do with the taking of a root'. For example, a radical equation is an equation containing a variable inside a radical, or the root of a variable.

A radical with no number to the left is used to represent a square root. For example, square root of 7. means the square root of 7. If an integer appears to the left of the radical sign, it means an nth root. The expression 3 radical sign 4. means the cube root of 4. The expression 4 radical sign 7. means the 4th root of 7.

A radicand is the number or expression inside a radical: radical(radicand). For example: in radical(3), 3 is the radicand; in 3 radical(x+2), x + 2 is the radicand; and in 3*4radical(t), t is the radicand.

Radicals and exponentiation

A radical is the same as a exponent of a unit fraction. For example, radical(3) = 3^(1/2). In general, a radical(x) = x^(1/a). To convert a radical to an exponent, make the radicand the base, and the number to the left of the radical the denominator of the unit fraction in the exponent.

a radical(x) = x^(1/a). Both a's are red. Both x's are blue. There is an arrow pointing from the a on the left to the a on the right. There is an arrow pointing from the x on the left to the x on the right.
Figure 1: Converting a radical to an exponent.

Simplifying radicals

A radical can be simplified if it contains a factor raised to the power of the radical. For example, Since radical(4) = radical(2^2), and radical(2^2) = 2, then radical(4) = 2. The steps to simplify a radical are:

  1. Find the prime factorization of the radicand. A factor tree is useful here.
  2. Divide into groups of factors with the same value. The size of each group is the number to the left of the radical or, if there is no number to the left of the radical, the size of the groups is 2.
  3. Bring out the groups of factors. For each group brought out, there is an exponent of 1. For example, in 3 radical(16*x^3*y^4), bring out factors in in groups of 3:
    3 radical(16*x^3*y^4) = 3 radical(2^4*x^3*y^4) = 3 radical(2^3*2*x^3*y^3*y).

Examples
Equation to simplifyPrime factorizationDivided into groupsGroups pulled out
radical(12x^2) radical(2^2*3*x^2) radical(underline(2^2)*3*underline(x^2)) 2*x*radical(3)
3 radical(x^7) 3 radical(x^3x^3x) 3 radical(underline(x^3)*underline(x^3)*x) x*x* 3 radical(x) = x^2 3 radical(x)
3 radical(12x) 3 radical(2^2*3*x) 3 radical(2^2*3*x). Note that there are no groups of 3. 3 radical(2^2*3*x). This radical can not be simplified.
4 radical(48x^7y^3) 4 radical(2^4*3*x^7y^3) 4 radical(underline(2^4)*3*underline(x^4)x^3y^3) 2x * 4 radical(3x^3y^3)
radical(x^3+x^2-x-1) radical((x+1)^2(x-1)) radical(Underline((x+1)^2)(x-1)) (x=1) radical(x-1)
Table 2: Examples of simplifying radicals

Common phrases using radical

  • radical expression: An expression containing a variable in a radical, such as 3+ radical( x + 2). Note that an expression with a number inside a radical, but no variable, is not a radical expression. x + radical(2) is not a radical expression.
  • radical equation: An equation containing a variable in a radical, such as  y = 4 - radical( x  - 1). Note that an equation with a number inside a radical, but no variable, is not a radical equation. y = 2x + radical(13) is not a radical equation.
  • radical function: A function containing a variable in a radical, such as f(x) = radical( 2x ). Note that a function with a number inside a radical, but no variable, is not a radical function. f(x) = radical(5)x is not a radical function.

Cite this article as:


Radical. 2010-12-06. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/radical.html.

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2010-12-06: Initial version (McAdams, David.)

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