Distributive law III
We've already learnt how to simplify expressions with grouping symbols. To expand an expression like $3\left(x+2\right)$3(x+2) or $5\left(2y-1\right)$5(2y−1) we use the distributive law:
To expand an expression of the form $A\left(B+C\right)$A(B+C), we use the property:
| $A\left(B+C\right)$A(B+C) | $=$= | $A\times B+A\times C$A×B+A×C |
| $=$= | $AB+AC$AB+AC |
So far we have used the distributive law to simplify expressions involving multiplication of constants with variables. Now we will look at how to use the distributive law to simplify expressions involving multiplication of variables. We will need to use the multiplication index law.
Example 1
Expand: $5x\left(6x^6-3y\right)$5x(6x6−3y)?
Think: We'll expand the brackets using the distributive law:
To evaluate the multiplications $5x\times6x^6$5x×6x6 and $5x\times\left(-3y\right)$5x×(−3y), we will use the power rule:
To multiply like terms with like bases, (e.g. $x$x and $x$x) we use the rule:
| $x^a\times x^b$xa×xb | $=$= | $x^{a+b}$xa+b |
For example,
| $x\times x^2$x×x2 | $=$= | $x^{1+2}$x1+2 |
| $=$= | $x^3$x3 |
Therefore:
| $5x\times6x^6$5x×6x6 | $=$= | $30x^7$30x7 |
| $5x\times\left(-3y\right)$5x×(−3y) | $=$= | $-15xy$−15xy |
Do: $30x^7-15xy$30x7−15xy
Examples
Question 1
Expand the following:
$r\left(r+5\right)$r(r+5)
Question 2
Question 3
Expand $6u^7\left(9u^7+9u^6\right)$6u7(9u7+9u6)
Question 4
Expand:
$7wy\left(y+w\right)$7wy(y+w)