Sure, let's simplify each of these expressions step by step using exponent properties.
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(x³y⁴)(x²y³)
- Combine like bases by adding their exponents: \[ x^{3+2}y^{4+3} = x^5y^7 \]
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(5y²w⁴)²(yw²)⁴
- First simplify each part: \[ (5^2y^{2 \cdot 2}w^{4 \cdot 2})(y^1w^{2 \cdot 4}) = (25y^4w^8)(y^4w^8) \]
- Combine like terms: \[ 25y^{4+4}w^{8+8} = 25y^8w^{16} \]
-
y⁹/y⁶
- Use the property of exponents for division: \[ y^{9-6} = y^3 \]
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r⁶n⁻⁷/r⁴n²
- Divide the bases by subtracting the exponents: \[ r^{6-4}n^{-7-2} = r^2n^{-9} = \frac{r^2}{n^9} \]
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(4x²y³)⁰/8x³
- Anything raised to the power of zero is 1 (except when it's zero itself): \[ \frac{1}{8x^3} \]
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15x³y⁵/18x⁵y²
- Simplify the coefficients and then apply the exponent properties: \[ \frac{15}{18} \cdot \frac{x^{3-5}}{y^{5-2}} = \frac{5}{6} \cdot \frac{1}{x^2} \cdot \frac{1}{y^3} = \frac{5}{6x^2y^3} \]
-
(5xy²z⁴/3xyz)³
- First simplify the fraction: \[ \frac{5}{3} \cdot \frac{x^{1}}{x^{1}} \cdot \frac{y^{2}}{y^{1}} \cdot \frac{z^{4}}{z^{1}} = \frac{5}{3} \cdot 1 \cdot y^{2-1} \cdot z^{4-1} = \frac{5}{3} y^1 z^3 = \frac{5yz^3}{3} \]
- Now raise the simplified fraction to the power of 3: \[ \left(\frac{5yz^3}{3}\right)^3 = \frac{5^3(y^3)(z^{9})}{3^3} = \frac{125y^3z^9}{27} \]
Here are the simplified expressions:
- \( x^5y^7 \)
- \( 25y^8w^{16} \)
- \( y^3 \)
- \( \frac{r^2}{n^9} \)
- \( \frac{1}{8x^3} \)
- \( \frac{5}{6x^2y^3} \)
- \( \frac{125y^3z^9}{27} \)