Question
Simplify. \[ \left(2 x^{5} y^{-4}\right)^{3} \] Write your answer using only positive exponents. \( \square \)
Ask by Flynn French. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{8x^{15}}{y^{12}} \).
Solution
To simplify the expression (2x⁵y⁻⁴)³, follow these steps:
1. Apply the exponent 3 to each factor inside the parentheses:
(2)³ ⋅ (x⁵)³ ⋅ (y⁻⁴)³
2. Simplify each term:
(2)³ = 8
(x⁵)³ = x^(5×3) = x¹⁵
(y⁻⁴)³ = y^(–4×3) = y⁻¹²
3. Write the expression with positive exponents:
8x¹⁵y⁻¹² = 8x¹⁵ / y¹²
Thus, the simplified expression is:
8x¹⁵ / y¹²
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Simplify this solution Bonus Knowledge
To simplify \(\left(2 x^{5} y^{-4}\right)^{3}\), we can apply the power of a product rule, which states that \((a \cdot b)^n = a^n \cdot b^n\). So, we have: \[ \left(2\right)^{3} \cdot \left(x^{5}\right)^{3} \cdot \left(y^{-4}\right)^{3} = 8 \cdot x^{15} \cdot y^{-12} \] Next, to express the answer with only positive exponents, we rewrite \(y^{-12}\) as \(\frac{1}{y^{12}}\). Thus, we get: \[ \frac{8x^{15}}{y^{12}} \] So, the final simplified expression is: \[ \frac{8x^{15}}{y^{12}} \]
