Multiplying a Monomial by a Monomial

Multiplying a Monomial by a Monomial: When a monomial is multiplied by another monomial the resulting product will also be a monomial. A monomial is an expression that has only one term in it. For example, x, y, 2x, 2y, x², y², etc. are monomials. A monomial cannot have negative exponents.

Now if we multiply a monomial by a monomial the resultant is a monomial. The coefficients of the monomials are multiplied together and then the variables are multiplied. For example, the product of two monomials, say 2x and 2y is equal to 4xy. If both the monomials have the same variables with the same exponents, then we need to use laws of exponents.

Multiplication of two monomials

As we have already discussed in the introduction, when two monomials are multiplied together then the result will be a monomial only. Let us see some examples below:

Examples:

1. 3x × 3y

Expanding the terms.

⇒ 3 × x × 3 × y

Multiplying the coefficients and variables separately, we get;

⇒ (3 × 3) × (x × y)

⇒ 9 × xy

⇒ 9xy

2. 5x × 4y

Expanding the terms.

⇒ 5 × x × 4 × y

Multiplying the coefficients and variables separately, we get;

⇒ (5 × 4) × (x × y)

⇒ 20 × xy

⇒ 20xy

3. 4x × (–3y)

Expanding the terms.

⇒ 4 × x × (–3) × y

Multiplying the coefficients and variables separately, we get;

⇒ 4 × (–3) × x × y

⇒ –12xy

Multiplication of Three or More Monomials

If we have to find the product of more than two monomials, then how can we do it? The answer is simple, first multiply the first two monomials, then the product of these two should be multiplied by the third monomial. In the same way, we can keep multiplying any number of monomials. Let us understand with the help of examples.

Examples:

1. 2x × 5y × 7z

First, multiply the monomials 2x and 5y together.

⇒ (2x × 5y) × 7z

⇒ 10xy × 7z

Now multiply the product with the third monomial

⇒ 10 × x × y × 7 × z [Separating each term]

⇒ (10 × 7) × x × y × z [Multiplying the coefficients and variables separately]

⇒ 70 xyz

2. xy × 2x² y² × 3x³y³

First, multiply the monomials 2x and 5y together.

= (xy × 2x² y² ) × 3x³ y³

By laws of exponents, we know that, a^m × aⁿ = a^mn. Therefore,

= 2 x³ y³ × 3x³y³

Multiplying the product with the third monomial

= 2 × x³ × y³ × 3 × x³ × y³ [Separating each term]

= 2 × 3 × x³ × x³ × y³ × y³ [Multiplying the coefficients and variables separately]

= 6 × x⁶ × y³

= 6 x⁶ y⁶

Problems and Solutions

Q.1: Multiply x and 3x³y

Solution: Given, two monomials are x and 3x³y

The product of two monomials:

⇒ x × 3x³y

⇒ x × 3 × x³ × y [Separating each term]

⇒ 3 × x × x³ x y [Multiplying the coefficients and variables separately]

⇒ 3 × x⁴ × y [By laws of exponents, a^m × aⁿ = a^mn]

⇒ 3x⁴y

Q.2: Find the volume of a cuboid, having length = 3x, breadth = 4y and height = 5z.

Solution: Given the dimensions of the cuboid are:

Length = 3x

Breadth = 4y

Height = 5z

Let the volume be V.

By the formula of volume of cuboid, we know that;

Volume = Length × Breadth × Height

V = 3x × 4y × 5z

Multiply the first two monomials,

V = (3x × 4y) × 5z

V = 12xy × 5z

V = 12 × 5 × x × y × z

V = 60 xyz cubic units.

Therefore, the required volume is 60 xyz cu.units.

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