Question

which of the following expressions is equivalent to $3 \cdot 147 - 3 \cdot 12$?
$3^2 \cdot 7^2 - 2^2 \cdot 3$
$3^2 \cdot 7^2 - 2^2 \cdot 3^2$
$3 \cdot 7^2 - 2^2 \cdot 3^2$
$3^3 \cdot 7^2 - 2^2 \cdot 3^2$
skill code: 104300

Explanation:

Step1: Factor the original expression

The original expression is \(3 \cdot 147 - 3 \cdot 12\). First, factor out the common factor \(3\): \(3(147 - 12)\). But we can also factor \(147\) and \(12\) into prime factors. \(147 = 3\times7^{2}\) and \(12 = 2^{2}\times3\). So the original expression becomes \(3\times(3\times7^{2})-3\times(2^{2}\times3)\)? Wait, no, let's do it step by step.

Wait, actually, \(147 = 3\times49=3\times7^{2}\), and \(12 = 3\times4 = 3\times2^{2}\). So the original expression \(3\cdot147 - 3\cdot12\) can be rewritten by substituting \(147 = 3\times7^{2}\) and \(12 = 3\times2^{2}\)? Wait, no, \(3\cdot147 = 3\times(3\times7^{2})=3^{2}\times7^{2}\)? Wait, no, \(3\times147 = 3\times(3\times49)=3\times3\times49 = 3^{2}\times7^{2}\), and \(3\times12 = 3\times(3\times4)=3\times3\times4 = 3^{2}\times2^{2}\). Wait, no, that's not right. Wait, \(147 = 3\times49 = 3\times7^{2}\), so \(3\times147 = 3\times3\times7^{2}=3^{2}\times7^{2}\), and \(3\times12 = 3\times(2^{2}\times3)=3^{2}\times2^{2}\). Wait, but the original expression is \(3\times147 - 3\times12\), so substituting \(147 = 3\times7^{2}\) and \(12 = 2^{2}\times3\), we get \(3\times(3\times7^{2})-3\times(2^{2}\times3)\)? No, that's overcomplicating. Wait, let's check the options.

Wait, the third option is \(3\cdot7^{2}-2^{2}\cdot3^{2}\)? No, wait, let's compute each option.

Wait, original expression: \(3\times147 - 3\times12 = 3\times(147 - 12)=3\times135 = 405\).

Now let's compute each option:

First option: \(3^{2}\cdot7^{2}-2^{2}\cdot3 = 9\times49 - 4\times3 = 441 - 12 = 429
eq405\).

Second option: \(3^{2}\cdot7^{2}-2^{2}\cdot3^{2}=9\times49 - 4\times9 = 441 - 36 = 405\). Wait, that's equal. Wait, but let's check again.

Wait, original expression: \(3\times147 - 3\times12 = 3\times(147 - 12)=3\times135 = 405\).

Second option: \(3^{2}\times7^{2}-2^{2}\times3^{2}=9\times49 - 4\times9 = 441 - 36 = 405\). Yes, that's correct. Wait, but let's check the third option: \(3\times7^{2}-2^{2}\times3^{2}=3\times49 - 4\times9 = 147 - 36 = 111
eq405\). Fourth option: \(3^{3}\times7^{2}-2^{2}\times3^{2}=27\times49 - 4\times9 = 1323 - 36 = 1287
eq405\). First option: 429, second option: 405, which matches. Wait, but the checkmark is on the third option? Wait, maybe I made a mistake. Wait, let's re-express the original expression.

Original expression: \(3\cdot147 - 3\cdot12\). Factor out 3: \(3(147 - 12)=3\times135\). Now, 147 is \(3\times49 = 3\times7^{2}\), 12 is \(3\times4 = 3\times2^{2}\). So \(147 - 12 = 3\times7^{2}-3\times2^{2}=3(7^{2}-2^{2})\). Then the original expression is \(3\times3(7^{2}-2^{2})=3^{2}(7^{2}-2^{2})=3^{2}\times7^{2}-3^{2}\times2^{2}\), which is the second option: \(3^{2}\cdot7^{2}-2^{2}\cdot3^{2}\). So the correct option is the second one (the one with \(3^{2}\cdot7^{2}-2^{2}\cdot3^{2}\)).

Step2: Verify each option

• Option 1: \(3^{2}\cdot7^{2}-2^{2}\cdot3 = 9\times49 - 4\times3 = 441 - 12 = 429

eq405\)

• Option 2: \(3^{2}\cdot7^{2}-2^{2}\cdot3^{2}=9\times49 - 4\times9 = 441 - 36 = 405\) (matches)
• Option 3: \(3\cdot7^{2}-2^{2}\cdot3^{2}=3\times49 - 4\times9 = 147 - 36 = 111

eq405\)

• Option 4: \(3^{3}\cdot7^{2}-2^{2}\cdot3^{2}=27\times49 - 4\times9 = 1323 - 36 = 1287

eq405\)

So the correct option is the second one (the one with \(3^{2}\cdot7^{2}-2^{2}\cdot3^{2}\)).

Answer:

The correct option is the second one (the checkbox with \(3^{2}\cdot7^{2}-2^{2}\cdot3^{2}\))