# 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)(x

^{5}) = x

^{3 + 5} = x

^{8}
ii) (x²y)(x³y²) = x

^{2 + 3}y

^{1 + 2} = x

^{5}y³

**Further Examples**
iii) (6k²)(15k³) = 90k

^{2 + 3} = 90k

^{5}
iv) (-4m²n)(3m

^{4}n²) = [(-4)(3)](m

^{2 + 4})(n

^{1 + 2}) = -12m

^{6}n³

v) (-7p²q)(-5pq²) = [(-7)(-5)](p

^{2 + 1})(q

^{2 + 1}) = 35p³q³

vi) (3a²b³c)(-2b²c²d) = [(3)(-2)](a²)(b

^{3 + 2})(c

^{1 + 2})(d) = -6a²b

^{5}c³d

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) (a

³)

² = a

^{6}
[Multiply the exponents

3 x

2 =

6]

ii) (x

³y

²)

^{4} = x

^{12}y

^{8}
[

3 ×

4 =

12 ,

2 ×

4 =

8]

iii) (3x

³y

²)

³ = (3

^{1})

³(x

³)

³(y

²)

³ = 3³x

^{9}y

^{6} = 27x

^{9}y

^{6}
[1 × 3 = 3,

3 ×

3 =

9,

2 ×

3 =

6]

iv) (-7y

³k

^{4})

² = (-7

^{1})

²(y

³)

²(k

^{4})

² = 49y

^{6}k

^{8}
v) (-5c²d³)³ = (-5

^{1})³(c²)³(d³)³ = -125c

^{6}d

^{9}

## 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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