Asked by AI
Simplify the following fraction.
StartFraction 1 plus 4 x Superscript negative 1 Baseline minus 21 x Superscript negative 2 Over 1 minus 2 x Superscript negative 1 Baseline minus 63 x Superscript negative 2 EndFraction
StartFraction 1 plus 4 x Superscript negative 1 Baseline minus 21 x Superscript negative 2 Over 1 minus 2 x Superscript negative 1 Baseline minus 63 x Superscript negative 2 EndFraction
Answers
Answered by
GPT-4o mini
To simplify the fraction
\[
\frac{1 + 4x^{-1} - 21x^{-2}}{1 - 2x^{-1} - 63x^{-2}},
\]
we start by rewriting the expression in the numerator and the denominator in terms of positive powers of \( x \).
First, we can express \( x^{-1} \) and \( x^{-2} \) as \( \frac{1}{x} \) and \( \frac{1}{x^2} \) respectively.
1. **Rewriting the numerator:**
\[
1 + 4x^{-1} - 21x^{-2} = 1 + \frac{4}{x} - \frac{21}{x^2} = \frac{x^2 + 4x - 21}{x^2}.
\]
2. **Rewriting the denominator:**
\[
1 - 2x^{-1} - 63x^{-2} = 1 - \frac{2}{x} - \frac{63}{x^2} = \frac{x^2 - 2x - 63}{x^2}.
\]
Now we can substitute these simplifications back into our fraction:
\[
\frac{\frac{x^2 + 4x - 21}{x^2}}{\frac{x^2 - 2x - 63}{x^2}}.
\]
When dividing by a fraction, we can multiply by its reciprocal:
\[
\frac{x^2 + 4x - 21}{x^2 - 2x - 63}.
\]
Next, we simplify \( x^2 + 4x - 21 \) and \( x^2 - 2x - 63 \) by factoring.
3. **Factoring the numerator:**
We can look to factor \( x^2 + 4x - 21 \):
\[
x^2 + 4x - 21 = (x + 7)(x - 3).
\]
4. **Factoring the denominator:**
Next, we factor \( x^2 - 2x - 63 \):
\[
x^2 - 2x - 63 = (x - 9)(x + 7).
\]
Thus, our simplified fraction now looks like this:
\[
\frac{(x + 7)(x - 3)}{(x - 9)(x + 7)}.
\]
Now, we can cancel the common factor \( (x + 7) \):
\[
\frac{x - 3}{x - 9},
\]
provided that \( x \neq -7 \) (which avoids division by zero).
Finally, the simplified form of the original expression is
\[
\frac{x - 3}{x - 9},
\]
where \( x \neq 9 \) and \( x \neq -7 \).
\[
\frac{1 + 4x^{-1} - 21x^{-2}}{1 - 2x^{-1} - 63x^{-2}},
\]
we start by rewriting the expression in the numerator and the denominator in terms of positive powers of \( x \).
First, we can express \( x^{-1} \) and \( x^{-2} \) as \( \frac{1}{x} \) and \( \frac{1}{x^2} \) respectively.
1. **Rewriting the numerator:**
\[
1 + 4x^{-1} - 21x^{-2} = 1 + \frac{4}{x} - \frac{21}{x^2} = \frac{x^2 + 4x - 21}{x^2}.
\]
2. **Rewriting the denominator:**
\[
1 - 2x^{-1} - 63x^{-2} = 1 - \frac{2}{x} - \frac{63}{x^2} = \frac{x^2 - 2x - 63}{x^2}.
\]
Now we can substitute these simplifications back into our fraction:
\[
\frac{\frac{x^2 + 4x - 21}{x^2}}{\frac{x^2 - 2x - 63}{x^2}}.
\]
When dividing by a fraction, we can multiply by its reciprocal:
\[
\frac{x^2 + 4x - 21}{x^2 - 2x - 63}.
\]
Next, we simplify \( x^2 + 4x - 21 \) and \( x^2 - 2x - 63 \) by factoring.
3. **Factoring the numerator:**
We can look to factor \( x^2 + 4x - 21 \):
\[
x^2 + 4x - 21 = (x + 7)(x - 3).
\]
4. **Factoring the denominator:**
Next, we factor \( x^2 - 2x - 63 \):
\[
x^2 - 2x - 63 = (x - 9)(x + 7).
\]
Thus, our simplified fraction now looks like this:
\[
\frac{(x + 7)(x - 3)}{(x - 9)(x + 7)}.
\]
Now, we can cancel the common factor \( (x + 7) \):
\[
\frac{x - 3}{x - 9},
\]
provided that \( x \neq -7 \) (which avoids division by zero).
Finally, the simplified form of the original expression is
\[
\frac{x - 3}{x - 9},
\]
where \( x \neq 9 \) and \( x \neq -7 \).