Final answer:
The associative property allows numbers to be regrouped when multiplying without changing the result. For the expression (9x^2)x40, we can group (9x40) and then multiply by x^2 to simplify to 360x^2. This property is also used with exponents, demonstrated with x^p x x^q = x^(p+q) and (x^a)^b = x^(a.b).
Step-by-step explanation:
The associative property in mathematics allows us to regroup numbers when adding or multiplying without changing the result. When we have an expression, such as (9x2) x 40, we can regroup the numbers without altering the product. Simplifying this expression involves first dealing with the parentheses.
To apply the associative property, we regroup the factors: (9 x 40) x x2. This does not affect the multiplication since multiplication is associative. We can then multiply 9 by 40, which equals to 360, and rewrite the expression: 360 x x2.
Finally, we multiply 360 by x2 to get 360x2, which is the simplified form of the given expression.
Considering another example, such as the rule xpxq = x(p+q), we can think of this as an exponent rule where we add the exponents when multiplying the same base. Applying this to 32 x 35, we get 32+5 which simplifies to 37.
Similarly, the rule (xa)b = xa.b demonstrates that when raising a power to another power, we multiply the exponents.