Factoring: Is (r+y)^3 the same as [(1+y/r)^3∗r^3] ?

[HappyS5]

Homework Statement

I am just refreshing my calculus and have a question. I have a weakness with factoring.

Relevant Equations

$$(r+y)^3$$

Is $(r+y)^3$ the same as $[(1+y/r)^3*r^3]$. If not, how do I factor r out of the parenthesis.

 

[member 587159]

Yes, that's correct, as long as $r \neq 0$. The $[]$ parentheses can be omitted though, they are unnecessary.

 

[Mark44]

Is $(r+y)^3$ the same as $[(1+y/r)^3*r^3]$. If not, how do I factor r out of the parenthesis.

$(r + y)^3 = \frac{r^3}{r^3}(r + y)^3 = r^3(\frac 1 {r^3}(r + y)^3) = r^3(\frac{r + y}r)^3 = r^3(1 + \frac y r)^3$.
As already noted, $r \ne 0$.

 

[member 587159]

$(r + y)^3 = \frac{r^3}{r^3}(r + y)^3 = r^3(\frac 1 {r^3}(r + y)^3) = r^3(\frac{r + y}r)^3 = r^3(1 + \frac y r)^3$.
As already noted, $r \ne 0$.

I did it this way:

$$(r+y)^3=(r(1+y/r))^3= r^3(1+y/r)^3$$

 

[PeroK]

Homework Statement:: I am just refreshing my calculus and have a question. I have a weakness with factoring.
Relevant Equations:: $$(r+y)^3$$

Is $(r+y)^3$ the same as $[(1+y/r)^3*r^3]$. If not, how do I factor r out of the parenthesis.

To see this holds for all natural exponents $n$, we can use the distributive law for the case $n = 1$:
$$(r + y) = (1 + \frac y r)r$$
And for the inductive step, also using the distributive law:
$$(r + y)^{n + 1} = (r+y)(r+y)^n = r(1 + \frac y r)r^n(1 + \frac y r)^n = r^{n +1}(1 + \frac y r)^{n + 1}$$