Question
Simplify. \[ \left(\frac{p^{-2}}{3 p^{-7}}\right)^{-3} \] Write your answer using only positive exponents.
Ask by Simpson Zimmerman. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \( \frac{27}{p^{15}} \).
Solution
Calculate or simplify the expression \( (p^{-2}/(3*p^{-7}))^{-3} \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(\frac{p^{-2}}{\left(3p^{-7}\right)}\right)^{-3}\)
- step1: Remove the parentheses:
\(\left(\frac{p^{-2}}{3p^{-7}}\right)^{-3}\)
- step2: Multiply by \(a^{-n}:\)
\(\left(\frac{p^{-2}\times p^{7}}{3}\right)^{-3}\)
- step3: Multiply the terms:
\(\left(\frac{p^{5}}{3}\right)^{-3}\)
- step4: Transform the expression:
\(\left(\frac{3}{p^{5}}\right)^{3}\)
- step5: Rewrite the expression:
\(\frac{3^{3}}{\left(p^{5}\right)^{3}}\)
- step6: Evaluate the power:
\(\frac{27}{p^{15}}\)
The simplified form of the expression \(\left(\frac{p^{-2}}{3 p^{-7}}\right)^{-3}\) is \(\frac{27}{p^{15}}\).
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Simplify this solution Bonus Knowledge
To simplify the expression \(\left(\frac{p^{-2}}{3 p^{-7}}\right)^{-3}\), we first handle the fraction inside the parentheses: \[ \frac{p^{-2}}{3 p^{-7}} = \frac{p^{-2}}{p^{-7}} \cdot \frac{1}{3} = \frac{p^{-2 + 7}}{3} = \frac{p^{5}}{3} \] Now we rewrite the expression: \[ \left(\frac{p^{5}}{3}\right)^{-3} \] Applying the power of a quotient rule, we can express this as: \[ \frac{(p^{5})^{-3}}{3^{-3}} = \frac{p^{-15}}{\frac{1}{27}} = p^{-15} \cdot 27 = 27p^{-15} \] Finally, to express the result using only positive exponents, we write: \[ \frac{27}{p^{15}} \] Thus, the simplified expression is: \[ \boxed{\frac{27}{p^{15}}} \]
