If I have this $(3 \times 4 + 2)^2$,
How can I simplify it with out the final result.
Do I distribute the $^2$ over each number like this:
$(3^2 \times 4^2 + 2^2)$?
What is the rule?
If I have this $(3 \times 4 + 2)^2$,
How can I simplify it with out the final result.
Do I distribute the $^2$ over each number like this:
$(3^2 \times 4^2 + 2^2)$?
What is the rule?
Can you simplify $(3\times4 + 2)?$
then$(3\times4 + 2)^2$
by definition $a^2 = a\times a$
let a=$(3\times4 + 2)$
$ = (3\times4 + 2)\times(3\times4 + 2)$ multiplication before addition within the parentheses
$ =(12 + 2) \times(12 + 2)$ $ =(14)\times(14)$ $= 196$
The distributive property of multiplication applies to coefficients not to exponents.
a(b+c) = ab+ac
If the 2 was in front as a coefficient then you could write:
$2(3\times 4 +2)$
$= 2\times3\times 4 +2\times 2$
The exponent sort of "distributes" when using the rule of exponents for a power of a product: $(ab)^m = a^mb^m$
No you can't distribute the powers like what you did.
There is a theorem called the binomial theorem that controls this type of operations.
The result is clearly $196$. Here is one way to get this result (steps are simplified so that you can follow)
You can do this to simplify the expression:
$x=(3\cdot4+2)^2$
then
$x=(12+2)(12+2)$
$x=(12\cdot12)+2\cdot(12\cdot2)+(2\cdot2)$
$x=144+48+4 = 196$
An expression like:
$x=(a+b)^2$
can be written as:
$x=(a+b)(a+b)=a\cdot a+2\cdot a\cdot b+a\cdot a = a^2+2ab+b^2$
$(3 \times 4 + 2)^2 = (12 + 2)^2 = 14^2 =196$ while $(3^2 \times 4^2 + 2^2) = 9 \times 16 +4 = 144+4 = 148$, so that does not work.
If you want a rule for squares of sums, try: $(x+y)^2 = x^2 + 2 x y +y^2.$
Always go basic by using the order of operations:
Using the PEMDAS rule, first simplify the parentheses, then simplify the exponent(s). We have,$ 3 \times 4 + 2$in the parentheses. Notice that again, PEMDAS is applied. Multiplication is done before addition. So, the simplification of the parentheses is as follows. $\begin{align}3 \times 4 + 2 & = \color{maroon}{3 \times 4} + 2 \\ & = 12 + 2 \\ & = 14 \end{align}$Now, the exponent. We'd have everything simplified as shown below: $\begin{align} (3 \times 4 + 2)^2 & = & 14^2 \\ & = & 14 \times 14 \\ & = & 196 \end{align} $
No, you can't distribute the power on any operator, power is distributed on $\times$ and $\div$ not on $+$ and $-$