Given the expression: $b^8(2b)^4$, simplify it.
I multiply $b^8$ and $2b$, which gives $(2b^9)^4.$
$(2b^9)^4 = 16b^{36}$. However this is incorrect.
The correct answer is to distribute the exponent first like so: $b^8(2^4b^4) = 16b^{12}$. This makes sense, but I'd like to know why my original approach was incorrect.
On
The point of the parentheses is that you are only applying the exponent ${}^4$ to the $2b$, inside the parentheses, not to the $b^8$ outside. Thus, we would apply the power to the parenthesis first, then evaluate the product.
On
If you pull the $b^8$ inside the parentheses, you need to reduce the exponent, because by putting it inside the parentheses, you are raising it to the $4th$ power: $$b^8(2b)^4 = (2b^3)^4 = 2^4b^{12} = 16b^{12}.$$ Alternatively, you could distribute the exponent to the $2b$ before multiplying in the $b^8$: $$b^8(2b)^4 = b^82^4b^4 = 2^4b^{12} = 16b^{12}.$$
On
$b^8(2b)^4$ is where you started. Then you moved the $b^8$ inside. However, to do this you have to compensate that inside the parens everything is raised to the 4th power, so you need to divide the exponent of $b$ by 4:
$b^8(2b)^4 = (b^2 \times 2b)^4 = (2b^3)^4$. Leaving the exponent unchanged when brought into the parentheses caused your error.
As an exercise: what is $r$ equal to in this equation: $x^9 = (x^r)^3$?
On
In my opinion, it's less that you violated a specific exponent property, more that you used one that doesn't exist.
The two most relevant exponent properties are that
$a^m \cdot a^n = a^{m + n}$; if you are multiplying two powers of the same base, you can add the exponents together. Or,
$a^n \cdot b^n = (a \cdot b)^n$; if you are multiplying two expressions raised to the same power, you can perform the multiplication first and then raise to that power (actually, on second thought, this is essentially the property you violated).
But $b^8(2b)^4$ is neither of those things: It's a product of $b^8$ and $(2b)^4$:
They don't have the same base ($b$ vs. $2b$), so the first property doesn't apply.
They're not being raised to the same exponent ($8$ vs $4$), so the second property doesn't apply.
You have violated the order of operations. $b^8 (2b)^4$ means that $2b$ is raised to the fourth power, then multiplied by $b^8$. If you were to perform multiplication of $b^8$ by $2b$ before you raise to the fourth power, you would also have raised $b^8$ to the fourth power, resulting in an additional factor of $b^{24}$. As you can see, $36 = 24 + 12$.
To illustrate how one might have written the expression to obtain your erroneous result, we would write instead $(b^8 (2b))^4 = 16b^{36}$.