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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?

  • 3
    How did this got tagged under number theory I wonder.2012-03-12

5 Answers 5

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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: $(ab)^m = a^mb^m$

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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:

$x=(a+b)(a+b)=a\cdot a+2\cdot a\cdot b+a\cdot a = a^2+2ab+b^2$

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$(3 \times 4 + 2)^2 = (12 + 2)^2 = 14^2 =196$ 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: $(x+y)^2 = x^2 + 2 x y +y^2.$

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    From $(x+y)^2 = x^2 + 2 x y +y^2$, let $x=2s$ and $y=1$ to get $(2s)^2+2\times 2s \times 1 + (1)^2$ which you can easily tidy up. Note that $(2s)^2 = 2^2 \times s^2$.2012-03-12
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Always go basic by using the order of operations:

  • Parentheses.
  • Exponents.
  • Multiplication and division.(left to right)
  • Subtraction and addition.(no same order)

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} $

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No, you can't distribute the power on any operator, power is distributed on $\times$ and $\div$ not on $+$ and $-$