- published: 14 May 2016
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In mathematics, a cube root of a number, denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt[3]{x}
The cube root operation is not associative or distributive with addition or subtraction.
The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers, for example:
but
The cube roots of a number x are the numbers y which satisfy the equation
If x and y are real, then there is a unique solution and so the cube root of a real number is sometimes defined by this equation. If this definition is used, the cube root of a negative number is a negative number.
If x and y are allowed to be complex, then there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. This can lead to some interesting results.
For instance, the cube roots of the number one are:
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry). It is special by being a cuboid and a rhombohedron.
The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2Coxeter planes.
For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are
while the interior consists of all points (x0, x1, x2) with −1 < xi < 1.
For a cube of edge length a,
As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares and second powers.
In vascular plants, the root is the organ of a plant that typically lies below the surface of the soil. This is not always the case, however, since a root can also be aerial (growing above the ground) or aerating (growing up above the ground or especially above water). Furthermore, a stem normally occurring below ground is not exceptional either (see rhizome). So, it is better to define root as a part of a plant body that bears no leaves, and therefore also lacks nodes. There are also important internal structural differences between stems and roots.
The first root that comes from a plant is called the radicle. The four major functions of roots are 1) absorption of water and inorganic nutrients, 2) anchoring of the plant body to the ground, and supporting it, 3) storage of food and nutrients, 4) vegetative reproduction. In response to the concentration of nutrients, roots also synthesise cytokinin, which acts as a signal as to how fast the shoots can grow. Roots often function in storage of food and nutrients. The roots of most vascular plant species enter into symbiosis with certain fungi to form mycorrhizas, and a large range of other organisms including bacteria also closely associate with roots.
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[The Emcee plays a recording on a phonograph.]
[BOY SOPRANO]
The sun on the meadow is summery warm.
The stag in the forest runs free.
But gather together to greet the storm.
Tomorrow belongs to me.
The branch of the linden is leafy and green,
The Rhine gives its gold to the sea.
But somewhere a glory awaits unseen.
Tomorrow belongs to me.
The babe in his cradle is closing his eyes
The blossom embraces the bee.
But soon, says a whisper;
"Arise, arise,
Tomorrow belongs..."
[EMCEE (spoken)]