To find the product of the expression \( ( \frac{2}{3}g^2 - 6n )^2 \) using the area model, we can start by expressing it in a more familiar form.
We know that squaring a binomial can be done using the formula:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
In this case, let:
- \( a = \frac{2}{3}g^2 \)
- \( b = 6n \)
Now we compute \( a^2 \), \( 2ab \), and \( b^2 \):
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Calculate \( a^2 \): \[ a^2 = \left(\frac{2}{3}g^2\right)^2 = \frac{4}{9}g^4 \]
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Calculate \( b^2 \): \[ b^2 = (6n)^2 = 36n^2 \]
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Calculate \( 2ab \): \[ 2ab = 2 \cdot \frac{2}{3}g^2 \cdot 6n = 2 \cdot \frac{12}{3}g^2 \cdot n = 8g^2n \]
Now, substitute these values into the squared binomial formula:
\[ ( \frac{2}{3}g^2 - 6n )^2 = a^2 - 2ab + b^2 = \frac{4}{9}g^4 - 8g^2n + 36n^2 \]
Now, we can fill in the orange and blue boxes based on what they represent:
- The orange box will contain \( a^2 \), which is \( \frac{4}{9}g^4 \).
- The blue box will contain \( b^2 \), which is \( 36n^2 \).
Finally, we combine everything to write the product in standard form:
\[ ( \frac{2}{3}g^2 - 6n )^2 = \frac{4}{9}g^4 - 8g^2n + 36n^2 \]
So, to summarize:
- Partial product in orange box: \( \frac{4}{9}g^4 \)
- Partial product in blue box: \( 36n^2 \)
- Product in standard form: \( \frac{4}{9}g^4 - 8g^2n + 36n^2 \)
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