To perform and simplify the operation
[tex]\[ \frac{16 x^3}{5 y^9} \cdot \frac{x^3 y^7}{80 x y^2}, \][/tex]
we start by breaking this down step-by-step.
1. Multiply the numerators:
[tex]\[ 16 x^3 \times x^3 y^7. \][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[ x^3 \times x^3 = x^{3+3} = x^6. \][/tex]
- Include the [tex]\( y \)[/tex] term:
[tex]\[ 16 x^3 \times x^3 y^7 = 16 x^6 y^7. \][/tex]
2. Multiply the denominators:
[tex]\[ 5 y^9 \times 80 x y^2. \][/tex]
- Combine the [tex]\( y \)[/tex] terms:
[tex]\[ y^9 \times y^2 = y^{9+2} = y^{11}. \][/tex]
- Include the [tex]\( x \)[/tex] term:
[tex]\[ 5 \times 80 x = 400 x. \][/tex]
Therefore,
[tex]\[ 5 y^9 \times 80 x y^2 = 400 x y^{11}. \][/tex]
3. Combine the fractions:
[tex]\[ \frac{16 x^6 y^7}{400 x y^{11}}. \][/tex]
Now we simplify the fraction by canceling common factors.
- Simplify the constants:
[tex]\[ \frac{16}{400} = \frac{16 \div 16}{400 \div 16} = \frac{1}{25}. \][/tex]
- Simplify the [tex]\( x \)[/tex] terms:
[tex]\[ \frac{x^6}{x} = x^{6-1} = x^5. \][/tex]
- Simplify the [tex]\( y \)[/tex] terms:
[tex]\[ \frac{y^7}{y^{11}} = y^{7-11} = y^{-4}. \][/tex]
4. Combining the results:
The simplified form of the given expression is:
[tex]\[ \frac{1 \times x^5}{25 \times y^{-4}} = \frac{x^5}{25 y^{-4}}. \][/tex]
Since [tex]\( y^{-4} = \frac{1}{y^4} \)[/tex]:
[tex]\[ \frac{x^5}{25 y^{-4}} = \frac{x^5}{25} \times y^4 = \frac{x^5 \times y^4}{25}. \][/tex]
Finally, the expression simplified is:
[tex]\[ \frac{x^5 y^4}{25}. \][/tex]
Therefore, the simplified form of the given fraction is:
[tex]\[ \frac{x^5}{25 y^{-4}}, \][/tex]
or equivalently,
[tex]\[ \frac{x^5}{25 y^{-4}} = 0 \equals 0. \][/tex]
