Fully simplify.
start fraction, start fraction, 1, divided by, x, end fraction, plus, start fraction, 1, divided by, x, squared, end fraction, divided by, start fraction, 1, divided by, x, end fraction, minus, x, end fraction
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1
−x
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To simplify this expression, let's break it down step by step.
The given expression is:
((1/x) + (1/x^2)) / (1/x - x)
1. First, let's simplify the numerator.
((1/x) + (1/x^2))
= (1/x) * (x^2/x^2) + (1/x^2) * (x/x)
= (x^2+x) / (x^2)
2. Now, let's simplify the denominator.
(1/x - x)
= (1/x) - (x * x/x)
= 1/x - x^2/x
= (1 - x^3) / x
3. Finally, let's simplify the entire expression by dividing the numerator by the denominator.
((x^2+x) / (x^2)) / ((1 - x^3) / x)
= (x^2+x) * (x / (x^2 * (1 - x^3)))
= (x^3 + x^2) / (x^3 - x^5)
Therefore, the fully simplified form of the expression is:
(x^3 + x^2) / (x^3 - x^5)
To simplify this expression, we'll start by combining the two fractions inside the numerator:
[start fraction, 1, divided by, x, end fraction, plus, start fraction, 1, divided by, x squared, end fraction, divided by, start fraction, 1, divided by, x, end fraction]
To combine these fractions, we need to find a common denominator. The common denominator for x and x squared is x squared:
[start fraction, (1/x), multiplied by, (x squared/x squared), plus, (1/x squared), multiplied by, (x/x), end fraction, divided by, (1/x)]
Simplifying the numerators, we have:
[start fraction, (x squared/x squared), plus, (x/x squared), end fraction, divided by, (1/x)]
Simplifying further:
[start fraction, x squared/x squared, plus, x/x squared, end fraction, divided by, 1/x]
Now, we can divide the terms in the numerator by their common denominator:
[start fraction, (x squared + x)/x squared, end fraction, divided by, 1/x]
Dividing by a fraction is the same as multiplying by its reciprocal:
[start fraction, (x squared + x)/x squared, end fraction, multiplied by, x/1]
Multiplying the numerators and denominators, we get:
(start fraction x squared + x) over (x squared multiplied by 1) multiplied by x over 1
Simplifying further:
(x squared + x)/(x squared) multiplied by x/1
Simplifying the multiplication:
(x squared + x) times x/x squared = x squared + x squared = 2x squared.
So the fully simplified expression is 2x squared.
To fully simplify the expression, start by rationalizing the denominators within the fractions:
Rewriting the expression with rationalized denominators:
((1/x) + (1/x^2)) / ((1/x) - x)
To combine the fractions, we'll find a common denominator:
((1/x) * (x^2/x^2) + (1/x^2) * (x/x)) / ((1/x) * (x^2/x^2) - (x * (x^2/x^2)))
Simplifying further:
((x^2 + x) / x^3 + (x / x^2)) / ((x^2 - (x^3 / x^2)))
Combining like terms:
(x^2 + x + x) / x^3 / (x^2 - x)
Simplifying the expression:
(x^2 + 2x) / x(x^2 - x)
Therefore, the fully simplified expression is (x^2 + 2x) / x(x^2 - x).