Multiplication and division of algebraic expressions with whole

Multiplication and Division of Algebraic Expressions with Whole Number Coefficients

When multiplying or dividing the same variables, the governing rules can be deduced. Algebraic expressions with unlike variables are called unlike terms. Variables of unlike terms cannot be simplified when multiplied or divided.

Example 1

Simplify: $b \times b$.

Solution

$b \times b = b^1 \times b^1$

Since they have the same bases add exponents as follows:

$b \times b = b^{1 + 1}$

$= b^2.$

Therefore, $b \times b = b^2.$

Example 2

Simplify: $y^2 \times y^3$.

Solution

Since the bases are the same, add the exponents as follows:

$y^2 \times y^3 = (y^1 \times y^1) \times (y^1 \times y^1 \times y^1)$

$= y^{(1 + 1) + (1 + 1 + 1)}$

$= y^{2 + 3}$

$= y^5.$

Therefore, $y^2 \times y^3 = y^5.$

Using Examples 1 and 2, the following rule is deduced:

$a^m \times a^n = a^{m + n}$

Example 3

Simplify: $2m \times n \times 3p$.

Solution

Expand the given algebraic expression as follows:

$2m \times n \times 3p = 2 \times 3 \times m \times n \times p.$

Multiply the coefficients and simplify:

$2 \times 3 \times m \times n \times p = 6mnp.$

Therefore, $2m \times n \times 3p = 6mnp.$

Example 4

Simplify: $a^3 \div a^2$.

Solution

$a^3 \div a^2 = \frac{a^3}{a^2}$

$= \frac{a \times a \times a}{a \times a}$

$= a.$

Therefore, $a^3 \div a^2 = a.$

Using Example 4, the following rule is deduced:

$a^m \div a^n = a^{m - n} \quad \text{or} \quad \frac{a^m}{a^n} = a^{m - n}$

Example 5

Simplify: $7xy^3 \times 10x^4y$.

Solution

Expand the given algebraic expression as follows:

$7xy^3 \times 10x^4y = 7 \times x \times y^3 \times 10 \times x^4 \times y$

$= 7 \times 10 \times x \times y^3 \times x^4 \times y.$

Multiply the coefficients and variables:

$7 \times 10 \times x \times y^3 \times x^4 \times y = 70 \times x^{(1 + 4)} y^{(3 + 1)}$

$= 70x^5y^4.$

Therefore, $7xy^3 \times 10x^4y = 70x^5y^4.$

Example 6

Simplify: $4p^2t \div 2pt$.

Solution

Write the algebraic expression in fraction: $4p^2t \div 2pt = \frac{4p^2t}{2pt}$

Expand $\frac{4p^2t}{2pt}$ as follows:

$\frac{4p^2t}{2pt} = \frac{4 \times p \times p \times t}{2 \times p \times t}.$

Collect like terms:

$\frac{4 \times p \times p \times t}{2 \times p \times t} = \frac{4}{2} \times \frac{p^2}{p} \times \frac{t}{t}.$

Simplify the algebraic expression obtained:

$\frac{4}{2} \times \frac{p^2}{p} \times \frac{t}{t} = 2p.$

Therefore, $4p^2t \div 2pt = 2p.$