## Algebraic Expressions Involving Basic Arithmetic Operations

5.2  Algebraic Expressions Involving Basic Arithmetic Operations

Add and subtract two or more algebraic expressions:

1. Gather the like terms.
2. Add or subtract the like terms.

• When the ‘$$+$$’ sign which lies before the brackets is removed, the sign for each term in the brackets remains unchanged.
• When the ‘$$-$$’ sign which lies before the brackets is removed, the sign for each term in the brackets changes from: ‘$$+$$ to $$-$$’; ‘$$-$$ to $$+$$’.

 Example Solve $$(6p+2q+10)-(p+3q-2)$$. \begin{aligned}&\space(6p+2q+10)-(p+3q-2) \\\\&=6p+2q+10-p-3q+2 \\\\&=6p-p+2q-3q+10+2 \\\\&=5p-q+12. \end{aligned}

Generalisation about repeated multiplication of algebraic expressions:

• Repeated multiplication of the algebraic expression $$(a+b)$$ by $$n$$ times.

\begin{aligned} &\space(a+b)^n \\\\&=(a+b)\times(a+b)\times...\times(a+b) \end{aligned}

 Example \begin{aligned} &\space(h-4k)^3 \\\\&=(h-4k)\times(h-4k)\times(h-4k) \end{aligned} We can see that $$n=3$$ means the multiplication is repeated $$3$$ times.

Multiply and divide algebraic expressions with one term:

• To find the product of algebraic expressions with one term, gather all the same variables, and then multiply the number with number and the variable with variable.

 Example Calculate the product of $$-4mn\times7m^2$$. \begin{aligned}&\space-4mn\times7m^2 \\\\&=-4\times m\times n\times7\times m\times m \\\\&=-4\times7\times m\times m\times m\times n \\\\&=-28m^3n. \end{aligned}

• The quotient of algebraic expressions with one term can be obtained by eliminating the common factors.

 Example Calculate the quotient of $$36pq\div4q^2$$. \begin{aligned}&\space36pq\div4q^2 \\\\&=\dfrac{36pq}{4q^2} \\\\&=\dfrac{36\times p\times q}{4\times q\times q} \\\\&=\dfrac{9p}{q}. \end{aligned}