Monomial

A monomial is an algebraic expression which consists of one term only.




Examples
9x, 7a², -3mpx²
The number in front of the variables (letters) is called the coefficient. In the above 3 examples, 9, 7, and -3 are the coefficients.


Addition & Subtraction

In addition & subtraction, add or subtract the coefficients only and leave the variables intact (the same).

Examples
i) 15ab – 18ab = -3ab
ii) 3x + 2x – 7x = -2x
iii) -3y + 9y = 6y
iv) 17q + 9q – 4q – (-4q) = 26q – 4q + 4q = 26q


Multiplication of monomials


If 3.3 = 3² and y.y.y = y³, then a.a.a.b.b = a³b²

In multiplication, add the exponents of the same bases.

Examples
i) (x³)(x5) = x3 + 5 = x8
ii) (x²y)(x³y²) = x2 + 3y1 + 2 = x5

Further Examples
iii) (6k²)(15k³) = 90k2 + 3 = 90k5
iv) (-4m²n)(3m4n²) = [(-4)(3)](m2 + 4)(n1 + 2) = -12m6
v) (-7p²q)(-5pq²) = [(-7)(-5)](p2 + 1)(q2 + 1) = 35p³q³
vi) (3a²b³c)(-2b²c²d) = [(3)(-2)](a²)(b3 + 2)(c1 + 2)(d) = -6a²b5c³d

In the above 4 examples, multiply the coefficients as well as add the exponents of the same bases. When an algebraic term is being raised to a power, then multiply the exponents of each part of the term, (including the coefficient) by the power to which it is raised.

Examples
Multiply the exponents
i) (a³)² = a6
[Multiply the exponents 3 x 2 = 6]

ii) (x³y²)4 = x12y8
[3 × 4 = 12 , 2 × 4 = 8]

iii) (3x³y²)³ = (31)³(x³)³(y²)³ = 3³x9y6 = 27x9y6
[1 × 3 = 3, 3 × 3 = 9, 2 × 3 = 6]

iv) (-7y³k4)² = (-71)²(y³)²(k4)² = 49y6k8

v) (-5c²d³)³ = (-51)³(c²)³(d³)³ = -125c6d9


Division of monomials

In the division, subtract the exponents of the denominator or divisor from the exponents of the numerator or dividend of the same base.

Examples






In examples (v) – (vii), divide the coefficients as well.



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