Simplify the exponential expression. \( \left(\frac{-35 a^{5} b^{7}}{7 a^{7} b^{-2}}\right)^{3} \) \( \left(\frac{-35 a^{5} b^{7}}{7 a^{7} b^{-2}}\right)^{3}=\square \) (Simplify your answer. Use positive exponents only.)
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
The Deep Dive
First, let's simplify the expression inside the parentheses: \[ \frac{-35 a^{5} b^{7}}{7 a^{7} b^{-2}} = \frac{-35}{7} \cdot \frac{a^{5}}{a^{7}} \cdot \frac{b^{7}}{b^{-2}} \] Calculating each part: \[ \frac{-35}{7} = -5 \] \[ \frac{a^{5}}{a^{7}} = a^{5-7} = a^{-2} \] \[ \frac{b^{7}}{b^{-2}} = b^{7 - (-2)} = b^{7 + 2} = b^{9} \] Putting it all together, we have: \[ \frac{-35 a^{5} b^{7}}{7 a^{7} b^{-2}} = -5 a^{-2} b^{9} \] Next, we take this simplified form and raise it to the power of 3: \[ (-5 a^{-2} b^{9})^{3} = (-5)^{3} \cdot (a^{-2})^{3} \cdot (b^{9})^{3} \] Calculating each part: \[ (-5)^{3} = -125 \] \[ (a^{-2})^{3} = a^{-6} \] \[ (b^{9})^{3} = b^{27} \] Combining these, we get: \[ (-125) a^{-6} b^{27} \] To express with positive exponents, we rewrite \(a^{-6}\): \[ -125 \frac{b^{27}}{a^{6}} \] Thus, the simplified expression is: \[ \left(\frac{-35 a^{5} b^{7}}{7 a^{7} b^{-2}}\right)^{3} = -\frac{125 b^{27}}{a^{6}} \]
