1.02 Multiplying and dividing terms
We multiply and divide algebraic terms using this process:
- Split each term into its coefficient and its pronumerals.
- Find the product or quotient of the coefficient of the terms.
-
- When multiplying, combine like factors into a power. For example, $x\times x=x^2$x×x=x2.
- When dividing, cancel any common factors. For example, $x\div x=1$x÷x=1.
- Combine the coefficient and pronumerals into one term.
Unlike adding and subtracting, when we multiply or divide algebraic terms, we can collect them into one term.
Worked examples
Example 1
Simplify $6x\times9y$6x×9y.
Think: Here we have a product of algebraic terms, so we can follow the process above to simplify this.
Do: $6x$6x has a coefficient of $6$6 and a pronumeral $x$x. $9y$9y has a coefficient $9$9 and a pronumeral $y$y.
We first want to evaluate the product of the coefficients. Here we have $6\times9=54$6×9=54.
Next we look at the pronumerals in each term. $6x$6x has $x$x but not $y$y and $9y$9y has $y$y but not $x$x. So we cannot simplify the pronumerals any further.
This leaves us with the factors $54$54, $x$x, and $y$y. We can simplify this without writing the multiplication signs to get $54xy$54xy.
Example 2
Simplify $6xz\div\left(9yz\right)$6xz÷(9yz).
Think: Here we have a quotient of algebraic terms, so we can follow the same process as above except that we divide instead of multiplying.
We can also write this division as the fraction $\frac{6xz}{9yz}$6xz9yz which will make the simplification easier.
Do: $6xz$6xz has a coefficient of $6$6 and the pronumerals $x$x and $z$z. $9yx$9yx has a coefficient $9$9 and the pronumerals $y$y and $z$z.
We first want to simplify the quotient of the coefficients. Here we have $\frac{6}{9}=\frac{2}{3}$69=23.
Next we simplify the pronumerals. If we take just the pronumeral part of the fraction above we get $\frac{xz}{yz}$xzyz. $z$z is common to both the numerator and the denominator so we can cancel out $z$z, but we can't cancel out $x$x or $y$y.
This leaves us with the factors $\frac{2}{3}$23 and $\frac{x}{y}$xy. We can simplify this into the fraction $\frac{2x}{3y}$2x3y.
We multiply and divide algebraic terms using this process:
- Split each term into its coefficient and its pronumerals.
- Find the product or quotient of the coefficient of the terms.
-
- When multiplying, combine like factors into a power. For example, $x\times x=x^2$x×x=x2.
- When dividing, cancel any common factors. For example, $x\div x=1$x÷x=1.
- Combine the coefficient and pronumerals into one term.
Unlike adding and subtracting, when we multiply or divide algebraic terms, we can collect them into one term.
Practice questions
Question 1
Simplify the expression $9\times m\times n\times8$9×m×n×8.
Question 2
Simplify the expression $6u^2\times7v^8$6u2×7v8.
Question 3
Simplify the expression $\frac{63pq}{9p}$63pq9p.