Inverse Variation

In an Inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases..

Given two variables x and y, the statements

y varies inversely as x,

y is inversely proportional to x.

all mean that y=kx, for some fixed nonzero real number k. The constant k is called the constant of variation or constant of proportionality.

Simple inverse variation problems consist in find a formula and verify what happen with a quantity when we change the other quantity of the problem.

Direct variation relations have the form y = kx where x and y are variables and k is a non-zero constant. When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation.

In simpler terms, that means if A is always twice as much as B, then they directly vary.

Above are different types of variations.

CIRCLE SEGMENT

A circle segment is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding the circle's center.

A closed interval corresponding to a finite portion of an infinite line is known as a Line Segment. Line segments are generally labeled with two letters corresponding to their endpoints, say A and B, and then written AB.

The segment of the circle is the region bounded by a chord and the arc subtended by the chord. Finding the circle segment area is very similar to working with the area of a sector.

Multiplying and Dividing Radicals

Our today’s topic is Multiplying and Dividing radicals.

Radicals involve the use of the radical sign (√). Sometimes these are called surd. The radical sign is generally taken to indicate the principal root of the radicand, although any radicand will have n different nth roots. The term radical is sometimes used loosely to refer to the entire expression consisting of radical sign and radicand. A radical is a symbol for the indicated root of a number, for example a square root or cube root; the term is also synonymous for the root itself.

The expression &NA; = P is called the radical expression, where n is the indicated root index, R is a real number and P is the nth root of number R such that Pⁿ = R.

How to Simplify Radical Expression:

You can simplify radical expression by involving variables and also numbers. When using radical expressions you will be able to break down the number into smaller pieces.

The product of two radicals with same index n can be found by multiplying the radicands and placing the result under the same radical. For example, √9 × √25 = √(9×25) = √225 = 15, which is equal to 3 × 5 = √9 × √25. Similarly, radicals with the same index sign can be divided by placing the quotient of the radicands under the same radical, then taking the appropriate root.

Example problem on Radical Expressions Calculator:-

Solve the radical expression √30

Solution:

Given that √40

√40 written as √4 * 10 = 40

√40 = √4 * 10

= √2² * 10

4 has multiples of 2*2 =4

= √2² √10

= 2 √10

The solution is 2√ 10

ACUTE DEFINITION

An angle is the figure formed by two rays sharing a common end point, called the vertex of the angle. The size of an angle is normally characterized by the smallest positive rotation that maps one of the rays into the other.


There are different types of angles like, acute angle, obtuse angle, Right angle, Straight angle, reflex angle, adjacent angle etc.

An angle whose measure is less than 90° is known as an acute angle. It is also known as a positive angle that measures less than 90°

Acute Definition: The measure of an angle with a measure between 0° and 90° or with less than 90° radians.

To learn about special angle values click here.

PROPERTIES OF MULTIPLICATION

Multiplication (symbol "×") is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic. For example: 2 times 4 is written as 4*2=8
There are four properties of Multiplication which makes multiplication easier and simpler. They are as follows:

Commutative property: When two number are multiplied together, the product is the same regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4
Associative Property: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4)

Multiplicative Identity Property of Multiplication: The product of any number and one is that number.

Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3

Multiplication is often written using the multiplication sign "×" between the terms. It is also denoted by a middle dot or a period.

COMPLEX NUMBERS

Complex numbers are an algebraic way of coding points in the ordinary Euclidean plane so that translation (shift of position) corresponds to the addition of complex numbers and both rescaling (enlargement or shrinking) and rotation correspond to multiplication of complex numbers. ...

DEFINITION OF COMPLEX NUMBERS

Complex Numbers definition: "A complex number is a number comprising a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. ...

Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applies mathematics and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Complex numbers are plotted on the complex plane, on which the real part is on the horizontal axis, and the imaginary part on the vertical axis.

Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal..

Multiplying complex numbers is accomplished in a manner similar to multiplying two binomials. You can use the FOIL process of multiplication, the distributive property, or your personal favorite means of multiplying.

For properties of Complex number click here.

RATIONAL EXPRESSION

A rational expression is an algebraic expression of the form P/Q, where P and Q are simpler expressions (usually polynomials), and the denominator Q is not zero. We can manipulate rational expressions in the same way that we manipulate fractions.

Simplifying Rational Expression Calculator

Thinking back to when you were dealing with whole number fractions one of the first things you did was simplify them: You "cancelled off" factors which were in common between the numerator and denominator. You could do this because dividing any number by itself gives you just "1", and you can ignore factors of "1".To simplify a rational expression, we first factor both the numerator and denominator completely then reduce the expression by cancelling common factors. A rational expression can be simplified or reduced to lowest terms if all common factors from the numerator and denominator are canceled.

Listed in the following picture is one solved example on Simplifying Rational Expressions Calculator:-