square root

Examples of square root in the following topics:

• 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.
  • If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401?
  • If the square root of a number is taken, the result is a number which when squared gives the first number.
  • Roots do not have to be square.
  • However, using a calculator can approximate the square root of a non-square number:$\sqrt {3} = 1.73205080757$

• Imaginary Numbers

  • There is no such value such that when squared it results in a negative value; we therefore classify roots of negative numbers as "imaginary."
  • A radical expression represents the root of a given quantity.
  • When the radicand (the value under the radical sign) is negative, the root of that value is said to be an imaginary number.
  • Specifically, the imaginary number, $i$, is defined as the square root of -1: thus, $i=\sqrt{-1}$.
  • We can write the square root of any negative number in terms of $i$.

• Radical Equations

  • When solving equations that involve radicals, begin by asking: is there an x under the square root?
  • If there is an $x$, or variable, under the square root, the problem must be approached differently.
  • Square both sides of the equation if the radical is a square root; Cube both sides if the radical is a cube root.
  • To undo the radical symbol (square root), square both sides of the equation.
  • $(\sqrt{6x-2})^2=(10)^2$,  squaring a square root leaves the expression under the square root symbol and $10$ squared is $100$

• Solving Problems with Radicals

  • Roots are written using a radical sign, and a number denoting which root to solve for.
  • When none is given, it is an implied square root.
  • Roots are written using a radical sign.
  • If there is no denotation, it is implied that you are finding the square root.
  • Otherwise, a number will appear denoting which root to solve for.

• The Discriminant

  • The discriminant $\Delta =b^2-4ac$ is the portion of the quadratic formula under the square root.
  • If ${\Delta}$ is positive, the square root in the quadratic formula is positive, and the solutions do not involve imaginary numbers.
  • If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:
  • If ${\Delta}$ is less than zero, the value under the square root in the quadratic formula is negative:
  • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.

• Introduction to Radicals

  • Roots are the inverse operation of exponentiation.
  • For example, the following is a radical expression that reverses the above solution, working backwards from 49 to its square root:
  • For now, it is important simplify to recognize the relationship between roots and exponents: if a root $r$ is defined as the $n \text{th}$ root of $x$, it is represented as
  • If the square root of a number $x$ is calculated, the result is a number that when squared (i.e., when raised to an exponent of 2) gives the original number $x$.
  • This is read as "the square root of 36" or "radical 36."

• Simplifying Radical Expressions

  • where $n$ is the degree of the root.
  • A root of degree 2 is called a square root and a root of degree 3, a cube root.
  • Roots of higher degrees are referred to using ordinal numbers, as in fourth root, twentieth root, etc.
  • First, look for a perfect square under the square root sign and remove it:
  • First, notice that there is a perfect square under the square root symbol, and pull that out: $\frac{\sqrt{16}\sqrt{x^7}}{\sqrt[4]{x^2}} = \frac{4 \cdot \sqrt{x^7}}{\sqrt[4]{x^2}} = 4 \cdot \frac{\sqrt{x^7}}{\sqrt[4]{x^2}}$

• Factoring Perfect Square Trinomials

  • When a trinomial is a perfect square, it can be factored into two equal binomials.
  • Note that if a binomial of the form $a+b$ is squared, the result has the following form: $(a+b)^2=(a+b)(a+b)=a^2+2ab+b^2.$ So both the first and last term are squares, and the middle term has factors of $2,$ $a$, and $b,$ where the latter are the square roots of the first and last term respectively.
  • For example, if the expression $2x+3$ were squared, we would obtain $(2x+3)(2x+3)=4x^2+12x+9.$ Note that the first term $4x^2$ is the square of $2x$ while the last term $9$ is the square of $3$, while the middle term is twice $2x\cdot3$.
  • Suppose you were trying to factor $x^2+8x+16.$ One can see that the first term is the square of $x$ while the last term is the square of $4$.
  • Since the first term is $3x$ squared, the last term is one squared, and the middle term is twice $3x\cdot 1$, this is a perfect square, and we can write:

• Completing the Square

  • The method of completing the square allows for the conversion to the form:
  • Once completing the square has been performed, the quadratic is easy to solve; because there is only one place where the variable $x$ is squared, the $(x-h)^2$ term can be isolated on one side of the equation, and then the square root of both sides can be taken.
  • This quadratic is not a perfect square.  
  • The closest perfect square is the square of $5$, which was determined by dividing the $b$ term (in this case $10$) by two and producing the square of the result.
  • Solve for the zeros of a quadratic function by completing the square

• Factoring a Difference of Squares

  • When a quadratic is a difference of squares, there is a helpful formula for factoring it.
  • Taking the square root of both sides of the equation gives the answer $x = \pm a$.
  • Using the difference of squares is just another way to think about solving the equation.
  • If you recognize the first term as the square of $x$ and the term after the minus sign as the square of $4$, you can then factor the expression as:
  • If we recognize the first term as the square of $4x^2$ and the term after the minus sign as the square of $3$, we can rewrite the equation as: