cube root

Examples of cube 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.
  • The cube root of a number ($\sqrt [ 3 ]{x}$ ) can also be taken.
  • The cube root is the number which, when cubed, or multiplied by itself and then multiplied by itself again, gives back the original number.
  • For example, the cube root of 8 is 2 because $2 \cdot 2\cdot 2=8$ , or $\sqrt[3]{8}=2$
  • Irrational numbers also appear when attempting to take cube roots or other roots.

• Introduction to Radicals

  • Roots are the inverse operation of exponentiation.
  • 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
  • The cube root of a number ($\sqrt [ 3 ]{x}$ ) can also be calculated.
  • The cube root of a value $x$ is the number that when cubed (i.e., when raised to an exponent of 3) yields the original number $x$.
  • For example, the cube root of 8 is 2 because $2^3 = 2 \cdot 2\cdot 2=8$.

• Simplifying Radical Expressions

  • Expressions that include roots are known as 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:

• 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 $100$

• Roots of Complex Numbers

• The Rule of Signs

  • Finding the negative roots is similar to finding the positive roots.
  • Therefore it has exactly one positive root.
  • where $n$ is the total number of roots in a polynomial, $p$ is the maximum number of positive roots, and $q$ is the maximum number of negative roots.
  • Now we look for negative roots.
  • There are 2 complex roots.

• Zeros of Polynomial Functions with Real Coefficients

  • These are often called the roots of the function.
  • There are many methods to find the roots of a function.
  • Therefore both $-1$ and $2$ are roots of the function.
  • Even though all polynomials have roots, not all roots are real numbers.
  • Some roots can be complex, but no matter how many of the roots are real or complex, there are always as many roots as there are powers in the function.

• The Discriminant

  • A root is the value of the $x$ coordinate where the function crosses the $x$-axis.
  • The number of roots of the function can be determined by the value of $\Delta$.
  • If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:
  • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.
  • Because Δ is greater than zero, the function has two distinct, real roots.

• The Fundamental Theorem of Algebra

  • where $n > 0$ and $c_n \not = 0$, has at least one complex root.
  • For example, the polynomial $x^2 + 1$ has $i$ as a root.
  • admits one complex root of multiplicity $4$, namely $x_0 = 0$, one complex root of multiplicity $3$, namely $x_1 = i$, and one complex root of multiplicity $1$, namely $x_2 = - \pi$.
  • where $f_1(x)$ is a non-zero polynomial of degree $n-1.$ So if the multiplicities of the roots of $f_1(x)$ add to $n-1$, the multiplicity of the roots of $f$ add to $n$.
  • Therefore, a polynomial of even degree admits an even number of real roots, and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity).

• 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 $i$.