To find the product of \((\frac{2}{3}g^2 - 6n)^2\) using the area model, we can start by recognizing that this expression is a perfect square trinomial of the form \((a - b)^2\), where \(a = \frac{2}{3}g^2\) and \(b = 6n\).
The expansion of \((a - b)^2\) is given by:
\[ a^2 - 2ab + b^2 \]
We will calculate each term step by step:
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Calculate \(a^2\): \[ \left(\frac{2}{3}g^2\right)^2 = \left(\frac{2}{3}\right)^2 (g^2)^2 = \frac{4}{9}g^4 \]
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Calculate \(2ab\): \[ 2 \cdot \frac{2}{3}g^2 \cdot 6n = 2 \cdot \frac{2 \cdot 6}{3}g^2 n = \frac{24}{3}g^2 n = 8g^2 n \]
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Calculate \(b^2\): \[ (6n)^2 = 36n^2 \]
Now, we can substitute these values into the expansion formula:
\[ \left(\frac{2}{3}g^2 - 6n\right)^2 = a^2 - 2ab + b^2 = \frac{4}{9}g^4 - 8g^2n + 36n^2 \]
Now, let's identify the operations for the orange and blue boxes in the area model:
- The orange box is likely where we place the product \(a^2\) which is \(\frac{4}{9}g^4\).
- The blue box represents the product \(b^2\) which is \(36n^2\).
Finally, writing the final product in standard form, we have:
\[ \frac{4}{9}g^4 - 8g^2n + 36n^2 \]
In summary:
- Orange box: \(\frac{4}{9}g^4\)
- Blue box: \(36n^2\)
- Full product in standard form: \(\frac{4}{9}g^4 - 8g^2n + 36n^2\)
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