To simplify the expression \((37)8 \left(3^{7}\right)^{8}(3 7)8\), let's break it down step by step.
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Understanding Each Component:
- The term \((37)8\) can be interpreted as \(37^8\).
- The term \((3^7)^8\) can be simplified using the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\). Therefore, \((3^7)^8 = 3^{7 \cdot 8} = 3^{56}\).
- The term \((3 7)8\) can also be interpreted as being multiplied by \(37^8\), i.e., \(3^8 \cdot 7^8\).
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Combining Terms:
- Now we can combine everything: \[ 37^8 \cdot 3^{56} \cdot 3^8 \cdot 7^8 \]
- We can combine the bases for \(3\): \[ 3^{56} \cdot 3^8 = 3^{56 + 8} = 3^{64} \]
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Revisiting the Expression:
- Now rewriting our expression gives: \[ 37^8 \cdot 3^{64} \cdot 7^8 \]
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Final Form:
- The expression can be rearranged to group the powers: \[ 3^{64} \cdot 37^8 \cdot 7^8 \]
- We can represent \(37^8 \cdot 7^8\) together using the property of exponentiation \(a^m b^m = (ab)^m\): \[ 37^8 \cdot 7^8 = (37 \cdot 7)^8 = 259^8 \text{ (since } 37 \times 7 = 259) \]
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Resulting Exponential Form:
- Finally, we combine everything: \[ 3^{64} \cdot 259^8 \]
Thus, in exponential form, the simplified expression is:
\[ 3^{64} \cdot 259^8 \]
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