Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{q^{12}}{125 \, p^6}
\]
Solution
1. Aplicamos la propiedad de la potencia de un producto, es decir, \( \left(abc\right)^n = a^n b^n c^n \). Entonces:
\[
\left(5 p^{2} q^{-4}\right)^{-3} = 5^{-3} \cdot \left(p^{2}\right)^{-3} \cdot \left(q^{-4}\right)^{-3}.
\]
2. Simplificamos cada término aplicando la regla de las potencias \( \left(x^a\right)^b = x^{ab} \):
\[
5^{-3}, \quad \left(p^{2}\right)^{-3} = p^{2(-3)} = p^{-6}, \quad \left(q^{-4}\right)^{-3} = q^{-4(-3)} = q^{12}.
\]
3. Sustituyendo, obtenemos:
\[
\left(5 p^{2} q^{-4}\right)^{-3} = 5^{-3} \, p^{-6} \, q^{12}.
\]
4. Para expresar la respuesta sin exponentes negativos, recordamos que \( a^{-n} = \frac{1}{a^n} \). Así, se tiene:
\[
5^{-3} = \frac{1}{5^3} = \frac{1}{125} \quad \text{y} \quad p^{-6} = \frac{1}{p^6}.
\]
5. Por lo tanto, la expresión se puede escribir como:
\[
\frac{q^{12}}{125 \, p^6}.
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
To simplify the expression \( \left(5 p^{2} q^{-4}\right)^{-3} \), we can apply the power of a product rule. This means we distribute the exponent of -3 to each part inside the parentheses: \[ = 5^{-3} \cdot (p^{2})^{-3} \cdot (q^{-4})^{-3} \] Calculating each part gives us: \[ = \frac{1}{5^{3}} \cdot p^{-6} \cdot q^{12} \] Remember, \( 5^3 = 125 \), so we can rewrite the expression: \[ = \frac{q^{12}}{125p^{6}} \] Thus, the simplified form is: \[ \frac{q^{12}}{125p^{6}} \]
