# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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COUNTING NUMBERS

Reference: Integers.

CRITICAL POINT

Reference: Stationary Point.

CUBE (ALGEBRA) 129

CROSS MULTIPLY

Reference: Linear Equation.

CROSS SECTION

This is the flat surface obtained when a plane cuts straight through a solid shape. A good analogy is that of a saw cutting through a block of wood. The “saw cut” can be made at any angle. This is illustrated in the figure, as the saw cuts through a solid block of wood.

Vertical plane, and the cross section is a rectangle

Sloping plane, and the cross section is a triangle

Diagonal plane, and the cross section is a rectangle

References: Conic Sections, Cuboid, Cylinder, Parallel, Prism, Pyramid, Solids.

CUBE (ALGEBRA)

We cube a number, a quantity, or terms in algebra by multiplying the number by itself three times. For example, we cube the number 4 by working out 4 x 4 x 4 = 64. We say that 4 cubed = 64, or the cube of 4 is 64. Using the exponent notation, the abbreviation for 4 x 4 x 4 is 43, and we can write 43 = 64.

Example 1. Find the volume of a cube that has sides of length 4 cm.

Solution. Write

Volume =4x4x4 Volume = length x width x height

= 43 Using the index notation

The volume of the cube is 64 cubic centimeters.

Numbers can be cubed using a scientific calculator and you are referred to the calculator handbook. Terms in algebra can also be cubed, as explained in the following examples.

130 CUBE (ALGEBRA)

Example 2. Find the cube of each of these terms: (a) 3w2, (b) —2xfy2.

Solution. In algebra it is common to use a dot to stand for x when multiplying terms. For term (a),

(3w2)3 = 3w2 • 3w2 • 3w2 Multiplying the term by itself three times

= 3 • 3 • 3 • w2 • w2 • w2 Arranging the numbers together, and also the

terms

= 21 w6 3 • 3 • 3 = 27, and adding the exponents when

multiplying

For term (b),

Multiplying the term by itself three times —2 x — 2 x — 2 = —8 and y2 ■ y2 ■ y2 = y6

The inverse of cubing is cube rooting, or taking the cube root. The cube root of a number n is the number which, when cubed, gives n. This statement is also true for a quantity or an expression in algebra. For example, the cube root of 64 is 4, because 43 = 64. This is abbreviated using the cube root notation: -^64 = 4. Cube roots can also be written using the fraction 1/3 as the exponent. The cube root of 64 can be written as 641/3 = 4. The cube root of a negative number is also negative. For example, the cube root of —64 is —4. Numbers can be cube rooted using a calculator, and once again you are referred to the calculator handbook.

Terms in algebra can also be cube rooted, as shown in the following examples.

Example 3. Find the cube root of each of the following: (a) x6, (b) 27/y3, (c)—8w6.

Solutions. For (a), write

The cube root of x6 = (x6)1/3 A cube root can be written as a power of 1/3.

= x2 6x^=2, and using a law of indices

Note. When finding a cube root, the exponent is divided by 3.

For (b),

27 3

The cube root of — = — Taking cube roots of the top line and the bottom line

y y

CUBE (GEOMETRY) 131

For (c),

The cube root of — 8w6 = —2w2 The cube root of —8 is —2, and the cube root

of w6 is w2

Example 4. If the volume of a cube is 216 cubic meters, find the length of one side of the cube.

Solution. The length of one side of the cube is the cube root of 216. Write

The length of one side = -^216

= 6 Using a calculator

The length of one side of the cube is 6 meters.

References: Algebra, Cube, Exponent, Hexomino, Inverse.

CUBE (GEOMETRY)

A cube is one of the five Platonic solids. It is also called a regular hexahedron, which means a solid with six identical plane faces. Each of its faces is a square; they are the same size, and meet at right angles. If the length of each side of the cube is x units, then the volume of the cube = x3 cubic units (see figure a).

/

/ /x

X

(a)

The surface area of the cube = 6x2, and is made up of the areas of six squares added together, where each square is of area x2.

The net of a cube, which is made up of six squares, can take many forms, and two of them are drawn in figure b.

(b)

132 CUBIC EQUATIONS

References: Congruent, Hexomino, Net, Platonic Solids, Right Angle, Square, Surface Area, Volume.

CUBE NUMBERS

These numbers are the cubes of the counting numbers:

Counting number 1 2 3 4 ... N

Cube number 13 = 1 23 = 8 3s = 27 43 = 64 ... N3

The first four cube numbers are {1, 8, 27, 64}.

Reference: Cube (algebra).

CUBE ROOT

Reference: Cube (algebra).

CUBIC EQUATIONS

A cubic equation is an equation in which 3 is the highest power to which the variable is raised. The following examples demonstrate how to solve a variety of cubic equations.

Example 1. Solve jc3 = -8.

Solution. Write

x = -\/^8 Cube root of — 8 = —2

x = — 2 By inspection, or calculator

Example 2. Solve 5jc3 = 4.

Solution. Write

x3 =0.8 Divide both sides of equation by 5

x =0.93 Cube root of 0.8 using calculator, to 2 dp

Example 3. Solve 5(x + 3)3 = 84.

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