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QuestionSimplify the fraction: t+3t−25−2t−1\frac{t + 3 t^{-2}}{5 - 2 t^{-1}}.

Studdy Solution

STEP 1

Assumptions1. The given complex fraction is t+3t−5−t−1\frac{t+3 t^{-}}{5- t^{-1}} . We need to simplify this fraction

STEP 2

To simplify this complex fraction, we first need to get rid of the negative exponents. We can do this by moving the terms with negative exponents from the numerator to the denominator and vice versa, and changing the sign of the exponent.
t+t−25−2t−1=t+t25−2t\frac{t+ t^{-2}}{5-2 t^{-1}} = \frac{t+\frac{}{t^2}}{5-\frac{2}{t}}

STEP 3

Next, we need to find a common denominator for the fractions in the numerator and the denominator. The common denominator is t2t^2.
t+3t25−2t=t3t2+3t25t2t2−2tt2\frac{t+\frac{3}{t^2}}{5-\frac{2}{t}} = \frac{\frac{t^3}{t^2}+\frac{3}{t^2}}{\frac{5t^2}{t^2}-\frac{2t}{t^2}}

STEP 4

implify the fractions in the numerator and the denominator.
t3t2+3t2t2t2−2tt2=t+3t2−2t\frac{\frac{t^3}{t^2}+\frac{3}{t^2}}{\frac{t^2}{t^2}-\frac{2t}{t^2}} = \frac{t+\frac{3}{t^2}}{-\frac{2}{t}}

STEP 5

Now, we can simplify the complex fraction by multiplying the numerator and the denominator by t2t^2.
t+3t25−2t=t3+35t2−2t\frac{t+\frac{3}{t^2}}{5-\frac{2}{t}} = \frac{t^3+3}{5t^2-2t} So, the simplified form of the complex fraction t+3t−25−2t−1\frac{t+3 t^{-2}}{5-2 t^{-1}} is t3+35t2−2t\frac{t^3+3}{5t^2-2t}.

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