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Answer :
To simplify the expression [tex]\( f^{-69} g^{91} \cdot f g^{-1} \cdot f^0 g^{17} \)[/tex] using positive exponents, follow these steps:
1. Combine the terms involving [tex]\( f \)[/tex]:
- Start with [tex]\( f^{-69} \)[/tex].
- From the second term [tex]\( f g^{-1} \)[/tex], add the exponent of [tex]\( f \)[/tex], which is 1.
- From the third term [tex]\( f^0 g^{17} \)[/tex], add the exponent of [tex]\( f \)[/tex], which is 0.
Therefore, the total exponent for [tex]\( f \)[/tex] is:
[tex]\[
-69 + 1 + 0 = -68
\][/tex]
2. Combine the terms involving [tex]\( g \)[/tex]:
- Start with [tex]\( g^{91} \)[/tex].
- From the second term [tex]\( f g^{-1} \)[/tex], add the exponent of [tex]\( g \)[/tex], which is -1.
- From the third term [tex]\( f^0 g^{17} \)[/tex], add the exponent of [tex]\( g \)[/tex], which is 17.
Therefore, the total exponent for [tex]\( g \)[/tex] is:
[tex]\[
91 - 1 + 17 = 107
\][/tex]
3. Express the final result using positive exponents:
The simplified expression is:
[tex]\[
f^{-68} g^{107}
\][/tex]
So, the result of simplifying the expression [tex]\( f^{-69} g^{91} \cdot f g^{-1} \cdot f^0 g^{17} \)[/tex] is [tex]\( f^{-68} g^{107} \)[/tex], using positive exponents.
1. Combine the terms involving [tex]\( f \)[/tex]:
- Start with [tex]\( f^{-69} \)[/tex].
- From the second term [tex]\( f g^{-1} \)[/tex], add the exponent of [tex]\( f \)[/tex], which is 1.
- From the third term [tex]\( f^0 g^{17} \)[/tex], add the exponent of [tex]\( f \)[/tex], which is 0.
Therefore, the total exponent for [tex]\( f \)[/tex] is:
[tex]\[
-69 + 1 + 0 = -68
\][/tex]
2. Combine the terms involving [tex]\( g \)[/tex]:
- Start with [tex]\( g^{91} \)[/tex].
- From the second term [tex]\( f g^{-1} \)[/tex], add the exponent of [tex]\( g \)[/tex], which is -1.
- From the third term [tex]\( f^0 g^{17} \)[/tex], add the exponent of [tex]\( g \)[/tex], which is 17.
Therefore, the total exponent for [tex]\( g \)[/tex] is:
[tex]\[
91 - 1 + 17 = 107
\][/tex]
3. Express the final result using positive exponents:
The simplified expression is:
[tex]\[
f^{-68} g^{107}
\][/tex]
So, the result of simplifying the expression [tex]\( f^{-69} g^{91} \cdot f g^{-1} \cdot f^0 g^{17} \)[/tex] is [tex]\( f^{-68} g^{107} \)[/tex], using positive exponents.
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