The given expression to simplify is
[tex] \frac{-4a^{-2}b^4}{8a^{-6}b^{-3}} , \text{Assuming that } a\neq 0, b\neq 0. [/tex]
We can Simplify the given rational expression using the following exponent property
[tex] \frac{1}{a^m}=a^{-m}, \frac{1}{a^{-m}}=a^m\\
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a^m*a^n=a^{m+n}\\
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\text{Now re-write the given expression as below}\\
\\ [/tex]
[tex] \frac{-4a^{-2}b^4}{8a^{-6}b^{-3}} =\frac{-4}{8} *\frac{a^{-2}}{a^{-6}} *\frac{b^4}{b^{-3}} \\
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\text{Now apply the exponent property and simplify}\\
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=\frac{-1}{2} *a^{-2}*a^6*b^4*b^3\\
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\text{Again apply the exponent property and simplify we get}\\
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=\frac{-1}{2} *a^{-2+6}*b^{4+3}\\
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=\frac{-1}{2} a^4b^7 [/tex]
Hence exponent of the variable b is 7
