# Multiplying Monomials

Multiplying monomials is a method of simplifying them by finding the product of their coefficients followed by the product of their variables.

For example, the product of two monomials, 5m and 7n, is:

5m ✕ 7n = (5 ✕ 7) ✕ (m ✕ n) = 35mn

Here, we observe that the product of monomials gives a monomial. Also, we usually write the variables in alphabetical order while writing the product.

Now, let us multiply another two monomials, 2p and 3p5.

On multiplying the coefficients, we get

(2 ✕ 3) = 6

On multiplying the variables, we get

(p ✕ p5), where both the variables have the same base and different exponents.

By applying the exponent rules (am ✕ an = am+n), we get

p1+5 = p6

Thus, the product is 6p6.

Find the product of 3a2b3 and (4ab)3

Solution:

3a2b3 ✕ (4ab)3
Applying the exponent rules (am ✕ bm = (ab)m), we get
3a2b3 ✕ 43a3b3 = 3a2b3 ✕ 64a3b3
Now, on multiplying the coefficients, we get
(3 ✕ 64) = 192
On multiplying the variables, we get
a2b3 ✕ a3b3 = (a2 ✕ a3) ✕ (b3 ✕ b3)
As we know, am ✕ an = am+n, the product of the variables is
a2+3b3+3
Thus, the product of the given monomials is obtained as 192a5b6

## With Three or More Monomials

To find the product of more than two monomials, we calculate the product of the first two monomials and then multiply it with the next monomial, and the process continues.

Let us multiply 3r4s3t, 2rs, and 5s2t3.

Now, on multiplying the coefficients, we get

(3 ✕ 2 ✕ 5) = 30

On multiplying the variables, we get

(r4s3t ✕ rs ✕ s2t3) = (r4 ✕ r) ✕ (s3 ✕ s ✕ s2) ✕ (t ✕ t3) = r1+4s3+1+2t1+3

Thus, the product is r5s6t4.

Multiply: (3x3)3, 2x2t, and 5x3t6.

Solution:

(3x3)3 ✕ 2x2t ✕ 5x3t6
Applying the exponent rules (am ✕ bm = (ab)m and (am)n = amn), we get
33(x3)3 ✕ 2x2t ✕ 5x3t6 = 27x9 ✕ 2x2t ✕ 5x3t6
Simplifying further, we get
(27 ✕ 2 ✕ 5) ✕ (x9 ✕ x2 ✕ x3) ✕ (t ✕ t6)
As we know, am ✕ an = am+n, the product of the given monomials is
270x9+2+3t1+6 = 270x14t7.