To evaluate \( \frac{12^2}{6^3} \), we start by calculating the powers:
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Calculate \( 12^2 \): \[ 12^2 = 144 \]
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Calculate \( 6^3 \): \[ 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216 \]
Now we can substitute these values back into the expression:
\[ \frac{12^2}{6^3} = \frac{144}{216} \]
Next, we will simplify the fraction \( \frac{144}{216} \). We can find the greatest common divisor (GCD) of 144 and 216. The factorization is as follows:
- \( 144 = 12 \times 12 = 2^4 \times 3^2 \)
- \( 216 = 6 \times 36 = 6 \times 6 \times 6 = 2^3 \times 3^3 \)
The GCD can be found by taking the minimum power of each prime factor:
- For \( 2 \): min(4, 3) = 3 → \( 2^3 \)
- For \( 3 \): min(2, 3) = 2 → \( 3^2 \)
So, \[ \text{GCD}(144, 216) = 2^3 \times 3^2 = 8 \times 9 = 72 \]
Now we divide both the numerator and denominator by their GCD:
\[ \frac{144 \div 72}{216 \div 72} = \frac{2}{3} \]
Thus, the simplified answer is:
\[ \frac{12^2}{6^3} = \frac{2}{3} \]
The final answer is: \[ \boxed{\frac{2}{3}} \]