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Answer
La expresión simplificada es \( m^{9} n^{3} p^{7^{3} \cdot 3} \).
Solution
Para simplificar la expresión \( \left(m^{3} n p^{7^{3}}\right)^{3} \), aplicamos la propiedad de las potencias que dice que \((a^m)^n = a^{m \cdot n}\).
1. Aplicamos la potencia a cada término dentro del paréntesis:
\[
\left(m^{3}\right)^{3} = m^{3 \cdot 3} = m^{9}
\]
\[
\left(n\right)^{3} = n^{1 \cdot 3} = n^{3}
\]
\[
\left(p^{7^{3}}\right)^{3} = p^{7^{3} \cdot 3} = p^{7^{3} \cdot 3}
\]
2. Ahora, combinamos todos los términos:
\[
\left(m^{3} n p^{7^{3}}\right)^{3} = m^{9} n^{3} p^{7^{3} \cdot 3}
\]
Por lo tanto, la expresión simplificada es:
\[
m^{9} n^{3} p^{7^{3} \cdot 3}
\]
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Simplify this solution Extra Insights
To simplify the expression \( \left(m^{3} n p^{7^{3}}\right)^{3} \), you want to use the power of a product property, which states that \((a \cdot b)^{c} = a^{c} \cdot b^{c}\). Applying this, we break it down into: 1. \( \left(m^{3}\right)^{3} = m^{3 \cdot 3} = m^{9} \) 2. \( \left(n\right)^{3} = n^{3} \) 3. \( \left(p^{7^{3}}\right)^{3} = p^{7^{3} \cdot 3} \) Now we need to simplify \( 7^{3} \cdot 3 = 343 \cdot 3 = 1029 \). Combining all parts, we get: \[ m^{9} n^{3} p^{1029} \] So, the fully simplified expression is: \[ m^{9} n^{3} p^{1029} \]
