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

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

$$(r+y)^3=(r(1+y/r))^3= r^3(1+y/r)^3$$
 
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  • #5
PeroK
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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:
$$(r + y)^{n + 1} = (r+y)(r+y)^n = r(1 + \frac y r)r^n(1 + \frac y r)^n = r^{n +1}(1 + \frac y r)^{n + 1}$$
 
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