radical

(noun)

A root (of a number or quantity).

Related Terms

  • extraneous solution
  • extraneous solutions
  • square
  • radicand
  • radical expression
  • root

Examples of radical in the following topics:

  • Solving Problems with Radicals

    • Roots are written using a radical sign, and a number denoting which root to solve for.
    • Roots are written using a radical sign.
    • Any expression containing a radical is called a radical expression.
    • You want to start by getting rid of the radical.
    • Do this by treating the radical as if it where a variable.
  • Adding, Subtracting, and Multiplying Radical Expressions

    • An expression with roots is called a radical expression.
    • To add radicals, the radicand (the number that is under the radical) must be the same for each radical, so, a generic equation will have the form:
    • Multiplication of radicals simply requires that we multiply the variable under the radical signs.
    • the value under the radical sign can be written as an exponent,
    • Then, the fraction under the radical sign can be addressed, and the radical in the numerator can again be simplified.
  • Fractions Involving Radicals

    • In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator.
    • When we are given expressions that involve radicals in the denominator, it makes it easier to evaluate the expression if we rewrite it in a way that the radical is no longer in the denominator.
    • You are given the fraction $\frac{10}{\sqrt{3}}$, and you want to simplify it by eliminating the radical from the denominator.
    • Recall that a radical multiplied by itself equals its radicand, or the value under the radical sign.
    • Therefore, multiply the top and bottom of the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, and watch how the radical expression disappears from the denominator:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$
  • Simplifying Radical Expressions

    • A radical expression that contains variables can often be simplified to a more basic expression, much as can expressions involving only integers.
    • Expressions that include roots are known as radical expressions.
    • A radical expression is said to be in simplified form if:
    • For example, let's write the radical expression $\sqrt { \frac { 32 }{ 5 } }$ in simplified form, we can proceed as follows.
    • This follows the same logic that we used above, when simplifying the radical expression with integers:
  • Radical Equations

    • If there is not an $x$ under the square root—if only numbers are under the radicals—the problem can be solved much the same way as if it had no radicals.
    • Steps to Solve a Radical Equation with a Variable Under the Radical
    • In this case, both sides must be squared to get rid of the radical.
    • Now, to undo the radical symbol (square root), square both sides of the equation (recall that squaring a square root removes the radical):
    • Solve a radical equation by squaring both sides of the equation
  • Introduction to Radicals

    • Radical expressions yield roots and are the inverse of exponential expressions.
    • Mathematical expressions with roots are called radical expressions and can be easily recognized because they contain a radical symbol ($\sqrt{}$).
    • For example, the following is a radical expression that reverses the above solution, working backwards from 49 to its square root:
    • In this expression, the symbol is known as the "radical," and the solution of 7 is called the "root."
    • This is read as "the square root of 36" or "radical 36."
  • Imaginary Numbers

    • A radical expression represents the root of a given quantity.
    • What does it mean, then, if the value under the radical is negative, such as in $\displaystyle \sqrt{-1}$?
    • When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number.
  • Domains of Rational and Radical Functions

    • Rational and radical expressions have restrictions on their domains which can be found algebraically or graphically.
    • To determine the domain of a radical function algebraically, find the values of $x$ for which the radicand is nonnegative (set it equal to $\geq 0$) and then solve for $x$.  
    • The radicand is the number or expression underneath the radical sign.  
    • Calculate the domain of a rational or radical function by finding the values for which it is undefined
  • Radical Functions

    • An expression with roots is called a radical function, there are many kinds of roots, square root and cube root being the most common.
    • An expression with roots is called a radical expression.
    • The shape of the radical graph will resemble the shape of the related exponent graph it were rotated 90-degrees clockwise.
    • Discover how to graph radical functions by examining the domain of the function
  • Rational Exponents

    • Rational exponents are another method for writing radicals and can be used to simplify expressions involving both exponents and roots.
    • For example, we can rewrite $\sqrt{\frac{13}{9}}$ as a fraction with two radicals:
    • Notice that the radical in the denominator is a perfect square and can therefore be rewritten as follows:
    • Relate rational exponents to radicals and the rules for manipulating them
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