Solving Monomials Online

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In mathematics, in the context of polynomials, the word monomial means one of two different things:

The first meaning is a product of powers of variables, or formally any value obtained from 1 by finitely many multiplications by a variable. If only a single variable x is considered this means that any monomial is either 1 or a power xn of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form xaybzc with a,b,c nonnegative integers (taking note that any exponent 0 makes the corresponding factor equal to 1).
The second meaning of monomial includes monomials in the first sense, but also allows multiplication by any constant, so that − 7x5 and (3 − 4i)x4yz13 are also considered to be monomials (the second example assuming polynomials in x, y, z over the complex numbers are considered).

(Source: From Wikipedia)

Rules for solving monomials - online

The following rules must be considered when solving monomials - online

Parentheses: Do arithmetic operations in parentheses first. If no ...
... parentheses, do it for the powers, multiply the second term, and then add.

Signs: Adding negative monomials is nothing but subtraction

Adding monomials: Add only like terms with same varibles, including same exponents.

Multiplying monomials: Multiply a monomial with every other terms all variables and add exponents with same base variables.

Dividing Monomials: Divide a monomial by the same variable and subtract the exponents with same base variables.

Examples for solving monomials - online

Additing monomials: If among a sum of monomials there are similar ones, the sum can be reduced to the more simple form:

Online example: a x 3 y 2 – 5 b 3 x 3 y 2 + c 5 x 3 y 2 = ( a – 5 b 3 + c 5 ) x 3 y 2 .

Multiplication of monomials: It can be solving by just if it has powers of the similar numerical coefficients. In this case the powers are Multiplication monomials can be solving by just if it has powers of the similar numerical coefficients. Added and then the numerical coefficients are multiplied.

Online example: 5 a x 3 z 8 (– 7 a 3 x 3 y 2) = – 35 a 4 x 6 y 2 z 8.

Division of monomials: In the divisional monomial the power of exponent in a divisor is subtracted from an exponent of the power in a dividend; an arithmetical coefficient of a dividend is divided by an arithmetical coefficient of a divisor.
Example:

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Online example: 65 a 4 x 3 z 9: 5 a x 3 z 6 = 9 a 3 x z 3.

In this branch of mathematics, we use letters similar to a, b, x and y to refer to numbers. Performing addition, subtraction, multiplication, division of roots on these signs and real numbers, we get what are known as the algebraic expressions. Signs in an algebraic expression are known as variables of the expression. For example, in ax + b, if a and b are particular numbers and x is not specified, then x is the variable of ax + b. In 2x2 + 3xy + y2, x and y are variables

Definition of monomial:

An algebraic expression of the form axn is known as a monomial in x where a is a known number, x is a variable and n is a non-negative integer. The number a is known the coefficient of xn and n, the degree of the monomial and a monomial is an only one term which contain no symbol plus (+) or minus (-).

For example, 5x4 is a monomial in x of degree 4 and 5 is the coefficient of x4.

Powers of monomial:

Apply the following set of laws to locate the powers of monomials. The similar set of laws can be used by means of negative exponents.

Power of a power: For a few number a and positive integers m and a (am)n = amn
Power of a product: For every numbers a and b and positive integer m, (ab)m = ambm.
Powers of a monomial: For every numbers a and b and positive integers m, n and p,

(ambn)p = amp bmp.

Examples of powers of monomial:

Let us see some examples of powers of monomial.

Example 1:

Simplify (x2)5

Solution:

( x2)5 = x2.3 (Power of a power)

= x6

Example 2:

Simplify (a2b3c) 2.

Solution.

(a2b3c) 2 = (a2)2(b3)2(c) 2 Power of a monomial

= a2.2 b3.2 c2

= a4b6c2

Pratice problem:

(i) (8ab) -3

(ii) (p3q)5

(iii) (-82)2

Answer key:

(i) `1/ (512a^3b^3)`

(ii) p15q3

(ii) 4096

These are examples of powers of monomial.

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