# Operations with Monomials

### Addition of Monomials

** Similar monomials** can be added.

The sum of the monomials is another monomial that has the same literal part and whose coefficient is the sum of the coefficients.

**ax ^{n} + bx^{n} = (a + b)x^{n}**

2x^{2}y^{3}z + 3x y^{3}z = 5x^{2}y^{3}z

If the monomials are not similar, a polynomial is obtained.

2x^{2}y^{3 } + 3x^{2}y^{3 }z

### Multiplication of a Number by a Monomial

The product of a number by a monomial is another similar monomial whose coefficient is the product of the coefficient of the monomial and the number.

5 · (2x^{2} y^{3 }z) = 10x^{2} y^{3 }z

### Multiplication of Monomials

The multiplication of monomials is another monomial that takes as its coefficient the product of the coefficients and whose literal part is obtained by multiplying the powers that have the same base.

**ax ^{n} · bx^{m} = (a · b)(x^{n} · x ^{ m}) **=

**(a · b)x**

^{n + m}(5x^{2} y^{3 }z) · (2 y^{2} z^{2}) = 10 x^{2} y^{5 }z^{3}

### Division of Monomials

Dividing monomials can only be performed if they have the same literal part and the degree of the dividend has to be greater than or equal to the corresponding divisor.

The division of monomials is another monomial whose coefficient is the quotient of the coefficients and its literal part is obtained by dividing the powers that have the same base.

ax^{n} : bx^{m} = (a : b) (x^{n} : x ^{ m}) = **(a : b)x ^{n − m}**

If the degree of divisor is greater, an algebraic fraction is obtained.

### Power of a Monomial

To determine the power of a monomial, every element in the monomial is raised to the exponent of the power.

**(ax ^{n})^{m} = a^{m} · x^{n · m}**

(2x^{3})^{3} = 2^{3 } ** · **(x^{3})^{3} = 8x^{9}

(−3x^{2})^{3} = (−3)^{3 }** · ** (x^{2})^{3} = −27x^{6}