Civil Engineering Online Review

1. Principle on Combination Analysis

“If a first event can transpire in n₁ ways and after this has happened a second event can transpire in n₂ ways, then the two events can happen in (n₁)(n₂) ways.” Note: This principle is extendible to more than two events.

2. Permutations

Permutation is a set of n elements arranged in a definite order.

2.1 Permutation of n different Elements Taken r at  a time ( r< n)

P _(n,r) = n ! / (n – r )!

Read more…

The terms in the expansion of a binomial to any real exponent possess properties/characteristics in sequential order that is convenient to utilize for the definition of the general term, and the development of useful formulas/relations.

Consider the following expansion:

( x ± y )⁰ = 1

( x ± y )¹ = x ± y

( x ± y )² = x ² ± xy + y ²

( x ± y )³ = x ³ ± 3x ²y + 3xy ² ± y ³

( x ± y )ⁿ = x ⁿ ± nx ^n-1y + [n ( n-1)x ^n-2 y ² ] / 2! ±                                                                               [ n ( n-1 ) (n-2 ) x ^n-3 y ³ ] / 3! + …..

Important Properties of  ( x ± y )ⁿ , with n a positive integral exponent:

Read more…

1. Arithmetic Progression ( A.P )

Definition: the sequence of terms a₁,a₂,a₃,……a _n-1, a _n such that

a₂ – a₁ = a₃ – a₂ = a _n – a _n-1 = d

where d is known as the common difference.

Read more…

1.The Linear Equation

General Form:  ax + b = 0

Solution (Root):   x = -b/a

2.The Quadratic Equation

General Form:   ax² + bx + c = 0

Reduced Form:  x² + px + q = 0

where:  p = b/a,  q = c/a

Read more…

1. Definition:

If b>0 and b≠1, and b^x = N, then x is the logarithms of N to the base “b” or      x = log _b N.

2. Properties of Logarithms

Read more…

1. Negative Integral Exponent

x ^-m = 1 / x ^m

k / x ^-m = k x ^m

(x/y) ^-m = ( y / x )^m

2. Radical Term; Power with Fractional Exponent

The operation involved in the extraction of the nth root of a number N is called involution. It is usually expressed by the symbol called a radical, or

ⁿ √ N is read the “nth root of then number N”, where N is known as the radicand and n  is called the index.

Read more…

1. Multiplication

a. Product of two like powers ( Power with the same Base )

x ^m(x ⁿ) = x ^{n + m}

b. Power of a product

( xy ) ^m = x ^m y ^m

c. Power of a power

( x ^m ) ⁿ = x ^mn

2. Division

a. Quotient of two like powers

( x ^m / x ⁿ ) = x ^m-n = 1 / x ^n-m;            [if  m = n,   x⁰ = 1]

b. Power of a quotient

( x/y ) ^m = x^m / y^m