I have the following question on distributive property. If I multiply $3$ to the given expression $\frac{1}{3}\pi r^2h$.
Question: $3\cdot\frac{1}{3}\pi r^2h$
Based on what I understand for distributive property, I should take $3$ and multiply by every single term in the expression and I should expect $3\cdot\frac{1}{3} + 3\pi + 3r^2 + 3h$
However i did a check against Microsoft Mathematics is return the following answer $\pi r^2h$
Why is it so?
On
The distribute property is applicated to the sum:
$3⋅\left(\dfrac{1}{3}+π+r^2+h\right)=\left(3\cdot\dfrac{1}{3}+3\cdot π+3\cdot r^2+3\cdot h\right)$
But with the product you can't do that, in fact you may use the associative property:
$3⋅\left(\dfrac{1}{3}πr^2 h\right)=\left(3⋅\dfrac{1}{3}\right)πr^2 h=πr^2 h$
The distributive property reflects the interaction of two different operations: $$a\cdot(b+c)=ab+ac$$shows multiplication distributing over addition.
If your expression involves the same operation - so everything is multiplied - the key property is the associative law $$(ab)c=a(bc)$$ or $$(a+b)+c=a+(b+c)$$ for addition. This allows the brackets to be removed.
The other law which applies quite frequently to a single operation is the commutative law, which allows the order of elements to be changed eg$$ab=ba \text{ or } a+b=b+a$$
Your expression involves only multiplication, so the distributive law does not apply. You can use the associative law to bracket the first two items together and multiply them first $$3\cdot \frac 13=1$$ and that is all the simplification you can do.