Homework 5 Monomials All Operations
Homework 5 Monomials All Operations
A monomial is a type of polynomial that has only one term. A term is a product of a constant and one or more variables with non-negative integer exponents. For example, 3xy and -5z are monomials, but x and x are not.
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We can perform four basic operations on monomials: addition, subtraction, multiplication, and division. In this article, we will review the rules and examples for each operation.
Addition and Subtraction of Monomials
To add or subtract monomials, we need to make sure that they are like terms. Like terms are terms that have the same variables with the same exponents. For example, 2xy and -7xy are like terms, but 2xy and 3xy are not.
To add or subtract like terms, we simply add or subtract their coefficients and keep the same variable part. For example:
2xy + 5xy = (2 + 5)xy = 7xy
-4z - 6z = (-4 - 6)z = -10z
3ab - ab = (3 - 1)ab = 2ab
If the terms are not like terms, we cannot add or subtract them. We can only write them together using the appropriate sign. For example:
xy + xy is not simplified further.
-3ab + 5ab is not simplified further.
x + y is not simplified further.
Multiplication of Monomials
To multiply monomials, we use the following rules:
To multiply the coefficients, we use the usual rules of arithmetic.
To multiply the variables, we use the product rule of exponents, which says that when we multiply two powers with the same base, we add their exponents. For example, xx = x.
If there are different variables, we write them in alphabetical order.
If there is no coefficient, we assume it is 1.
If there is no exponent, we assume it is 1.
If there is no variable, we write only the coefficient.
For example:
(2x)(-3y) = (-6)xy (multiply the coefficients and write the variables in alphabetical order)
(-4xy)(5xy) = (-20)xy = -20xy
(a)(b) = ab (assume the coefficients and exponents are 1)
(7)(-8) = -56 (write only the coefficient)
(x)(y) = xy (write only the variables)
(x)(x) = xx (use the product rule of exponents)
Division of Monomials
To divide monomials, we use the following rules:
To divide the coefficients, we use the usual rules of arithmetic.
To divide the variables, we use the quotient rule of exponents, which says that when we divide two powers with the same base, we subtract their exponents. For example, x / x = x.
If there are different variables, we write them in alphabetical order.
If there is no coefficient, we assume it is 1.
If there is no exponent, we assume it is 1.
If there is no variable, we write only the coefficient.
If the denominator is zero, we say that the division is undefined.
For example:
(6x) / (-3y) = (-2)x / y (divide the coefficients and write the variables in alphabetical order)
(-12xy) / (3xy) = (-4)xy = -4xy
(a) / (b) = a / b (assume the coefficients and exponents are 1)
(8) / (4) = 2 (write only the coefficient)
(x) / (y) = x / y (write only the variables)
(x) / (x) = xx / x (use the quotient rule of exponents)
(x) / (0) = undefined (the denominator is zero)
Summary
In this article, we have learned how to perform four basic operations on monomials: addition, subtraction, multiplication, and division. We have seen that to add or subtract monomials, we need to have like terms and then add or subtract their coefficients. To multiply or divide monomials, we need to multiply or divide their coefficients and use the product or quotient rule of exponents for their variables. We have also learned some important rules and conventions for writing monomials in a simplified form. I'm glad you want to learn more about monomials. Here are some more topics that you might find interesting: Power of a Monomial
Sometimes, we need to raise a monomial to a certain power. For example, what is (2xy)? To find the answer, we use the following rules:
To raise a monomial to a power, we raise both the coefficient and the variables to that power.
To raise a coefficient to a power, we use the usual rules of arithmetic.
To raise a variable to a power, we use the power rule of exponents, which says that when we raise a power to another power, we multiply their exponents. For example, (x) = x.
For example:
(2xy) = (2)(x)(y) = 4xy = 4xy
(-3ab) = (-3)(a)(b) = -27abb = -27ab
(x) = xx = x(1*4)x = x4x (assume the coefficient and exponent are 1)
(5) = 5 * 5 = 25 (write only the coefficient)
(y) = y-1y (write only the variable)
(0)nn = 0 (if n is positive, 0 raised to any power is 0)
(0)-n-n = undefined (if n is positive, 0 raised to a negative power is undefined)
Greatest Common Factor and Least Common Multiple of Monomials
Sometimes, we need to find the greatest common factor (GCF) or the least common multiple (LCM) of two or more monomials. The GCF is the largest monomial that divides evenly into all the given monomials. The LCM is the smallest monomial that is divisible by all the given monomials. To find the GCF or LCM of monomials, we use the following steps:
To find the GCF or LCM of the coefficients, we use the usual methods of finding GCF or LCM of numbers.
To find the GCF or LCM of the variables, we use the following rules:
The GCF of two or more variables with the same base is the variable with the same base and the smallest exponent. For example, the GCF of x3x and x5x is x3x.
The LCM of two or more variables with the same base is the variable with the same base and the largest exponent. For example, the LCM of x3x and x5x is x5x.
If there are different variables, we include them all in the GCF or LCM.
If there is no variable, we write only the coefficient.
If there is no common factor or multiple, we write 1.
We write the GCF or LCM in a simplified form.
For example:
The GCF of 12x2x and 18x3x is 6x2x (the GCF of 12 and 18 is 6, and the GCF of x2x and x3x is x2x).
The LCM of 12x2x and 18x3x is 36x3x (the LCM of 12 and 18 is 36, and the LCM of x2x and x3x is x3x).
The GCF of 15ab and 20ab is 5ab (the GCF of 15 and 20 is 5, and the GCF of a and a is a, and the GCF of b and b is b).
The LCM of 15ab and 20ab is 60ab (the LCM of 15 and 20 is 60, and the LCM of a and a is a, and the LCM of b and b is b).
The GCF of x and y is 1 (there is no common factor).
The LCM of x and y is xy (there is no common multiple).
The GCF of 8 and 12 is 4 (write only the coefficient).
The LCM of xy and xy is xy (write only the variable).
Simplifying Fractions with Monomials
Sometimes, we need to simplify fractions that have monomials in the numerator or denominator. For example, what is the simplest form of (6xx / -9y)? To simplify fractions with monomials, we use the following steps:
We find the GCF of the numerator and denominator.
We divide both the numerator and denominator by the GCF.
We write the fraction in a simplified form.
If the denominator is negative, we move the negative sign to the numerator.
If the numerator or denominator has only one term, we write it as a monomial.
If the numerator or denominator has more than one term, we write it as a polynomial.
If there is no variable in the numerator or denominator, we write only the coefficient.
If there is no coefficient in the numerator or denominator, we assume it is 1.
If there is no exponent in the numerator or denominator, we assume it is 1.
If there is no fraction, we write only the monomial or polynomial.
For example:
(6x / -9y) = (-6x / 9y) / (3 / 3) = (-2x / 3y) (the GCF of -6x and 9y is -3, so we divide both by -3)
(-12a / -4a) = (-12 / -4) / (a / a) = (3 / 1) / (1 / 1) = (3 / 1) = 3 (the GCF of -12a and -4a is -4a, so we divide both by -4a)
(x + y) / (x - y) = (x + y) / (x - y) / (1 / 1) = (x + y) / (x - y) (there is no common factor)
Simplifying Fractions with Monomials (continued)
In the previous part, we learned how to simplify fractions that have monomials in the