radical expression

Examples of radical expression in the following topics:

• Simplifying Radical Expressions

  • Radical expressions containing variables can be simplified to a basic expression in a similar way to those involving only integers.
  • Expressions that include roots are known as radical expressions.
  • A radical expression is said to be in simplified form if:
  • Radical expressions that contain variables are treated just as though they are integers when simplifying the expression.
  • As with numbers with rational exponents, these rules can be helpful in simplifying radical expressions with variables.

• 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.
  • For example, the radical expression $\displaystyle \sqrt{\frac{16}{3}}$ can be simplified by first removing the squared value from the numerator.
  • Explain the rules for calculating the sum, difference, and product of radical expressions

• 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."

• 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:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$

• 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.

• 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.

• Domains of Rational and Radical Functions

  • Rational and radical expressions have restrictions on their domains which can be found algebraically or graphically.
  • A rational expression is one which can be written as the ratio of two polynomial functions.
  • The radicand is the number or expression underneath the radical sign.  
  • Example 3:  What is the domain of the radical function: $f(x) = \sqrt {x-3} +4$?
  • 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.
  • Using algebra will show that not all of these expressions are functions and that knowing when an expression is a relation or a function allows certain types of assumptions to be made.
  • 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:
  • This expression can be rewritten using the rule for dividing numbers with rational exponents:
  • 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

• Radical Equations

  • Equations involving radicals are often solved by moving the radical to one side of the equation and then squaring both sides.
  • Steps to Solve a Radical Equation with a variable under the radical symbol
  • Suppose $a$ and $b$ are algebraic expressions.  
  • Make sure the radical is positive.
  • $(\sqrt{6x-2})^2=(10)^2$,  squaring a square root leaves the expression under the square root symbol and $10$ squared is $100$