Simplification of algebraic expressions with decimal coefficients

Simplification of algebraic expressions with decimal coefficients

Algebraic expressions with decimal coefficients can be simplified by addition, subtraction, multiplication and division of the coefficients. As it is done in simplification of algebraic expressions with whole numbers and fractional coefficients, terms with decimal coefficients which can be added or subtracted are like terms only. Also, like variables in a term can be multiplied or divided.

Addition and subtraction of algebraic expressions with decimal coefficients

When adding or subtracting algebraic expressions with decimal coefficients, consider the like variables in the given terms. For example, $0.6x + 0.3y$ and $0.6x - 0.3y$ cannot be simplified because the terms have unlike variables.

Example 1

Simplify: $0.3x + 0.32x$.

Solution

These terms have like variables. Therefore, the terms can be added. Add the coefficients using a vertical method by considering the place value of the digits:

  0 . 3 0
+ 0 . 3 2
  0 . 6 2

Multiply the sum of coefficients by the given variable.

$0.62 \times x = 0.62x$

Therefore, $0.3x + 0.32x = 0.62x$.

Example 2

Simplify: $23.4ab^2 - 5.2ab^2$.

Solution

Subtract the coefficients using a vertical method by considering the place value of the digits:

2 3 . 4
- 5 . 2
1 8 . 2

Multiply the sum of coefficients by the given variable:

$18.2 \times ab^2 = 18.2ab^2$

Therefore, $23.4ab^2 - 5.2ab^2 = 18.2ab^2$.

Multiplication and division of algebraic expressions with decimal coefficients

Recall that, decimal numbers are multiplied as whole numbers, decimal places are counted and placed in the answer. When multiplying algebraic expressions with decimal coefficients, multiply coefficients of the terms, and then multiply the variables using the rule of multiplication of exponents. Also, when dividing terms with decimal coefficients, start dividing the coefficients, and then divide the variables by using the rule of dividing exponents.

Example 1

Simplify: $0.2y^3 \times 0.5y^4$

Solution

Multiply the coefficients and the variables as shown:

$0.2y^3 \times 0.5y^4 = 0.2 \times y^3 \times 0.5 \times y^4$

$= 0.2 \times 0.5 \times y^3 \times y^4$

$= 0.1 \times y^{3+4}$

$= 0.1y^7$

Therefore, $0.2y^3 \times 0.5y^4 = 0.1y^7$.

Example 2

Simplify: $1.2xy^2 \times 0.3x^3y$

Solution

Multiply the coefficients and the variables as shown:

$1.2xy^2 \times 0.3x^3y = 1.2 \times x \times y^2 \times 0.3 \times x^3 \times y$

$= 1.2 \times 0.3 \times x \times x^3 \times y^2 \times y$

$= 0.36 \times x^4 \times y^3$

$= 0.36x^4y^3$

Therefore, $1.2xy^2 \times 0.3x^3y = 0.36x^4y^3$.

Example 3

Simplify: $6.9b^4 \div 2.3b^3$

Solution

$6.9b^4 \div 2.3b^3 = \frac{6.9b^4}{2.3b^3}$

$= \frac{6.9 \times b \times b \times b \times b}{2.3 \times b \times b \times b}$

$= 3b$

Alternative solution

$6.9b^4 \div 2.3b^3 = \frac{6.9}{2.3} \times \frac{b^4}{b^3}$

$= 3 \times b^{4-3}$

$= 3b$

Therefore, $6.9b^4 \div 2.3b^3 = 3b$.