Questions: Use the power rule and the power of a product or quotient rule to simplify the expression. [ (-7 a^3 b^7 c)^2 ]

Solution Steps

The expression is \(\left(-7 a^{3} b^{7} c\right)^{2}\). According to the power rule, each factor inside the parentheses is raised to the power of 2: \[ \left(-7\right)^{2} \cdot \left(a^{3}\right)^{2} \cdot \left(b^{7}\right)^{2} \cdot \left(c\right)^{2}. \]

Calculate the power for each term: \[ \left(-7\right)^{2} = 49, \] \[ \left(a^{3}\right)^{2} = a^{3 \cdot 2} = a^{6}, \] \[ \left(b^{7}\right)^{2} = b^{7 \cdot 2} = b^{14}, \] \[ \left(c\right)^{2} = c^{2}. \]

Multiply the simplified terms together: \[ 49 \cdot a^{6} \cdot b^{14} \cdot c^{2}. \]

The final simplified expression is: \[ 49 a^{6} b^{14} c^{2}. \]

Final Answer