Example Question - simplify algebraic fractions
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Algebraic Fraction Simplification
The image contains an algebraic fraction that needs to be simplified. The given expression is: x = (y + 3) / (y + 3) To simplify this expression, you can cancel out the matching terms in the numerator and the denominator because (y + 3) divided by (y + 3) is 1, as long as y is not equal to -3 (because if y were -3, the denominator would be zero and the expression would be undefined). Therefore, the simplified expression is: x = 1 As long as y does not equal -3, x will equal 1.
Simplifying Algebraic Fraction with Exponents
To simplify the expression, we start by simplifying both the numerator and the denominator separately. We can simplify by combining like terms and applying the laws of exponents. Given expression: \[ \frac{10x^{n+4} + 125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \] First, let's simplify the numerator and the denominator by factoring out the common x term: Numerator: \[ 10x^{n+4} + 125x^{n+2} = x^{n+2}(10x^2 + 125) \] Denominator: \[ 3x^{n+3} - 20x^{n+1} = x^{n+1}(3x^2 - 20) \] Now, rewrite the expression using the factored terms: \[ \frac{x^{n+2}(10x^2 + 125)}{x^{n+1}(3x^2 - 20)} \] Next, cancel out the common x term from both the numerator and the denominator: since we have \(x^{n+2}\) in the numerator and \(x^{n+1}\) in the denominator, we can divide each by \(x^{n+1}\) so that: \[ x^{n+2} / x^{n+1} = x \] The simplified expression will be: \[ \frac{x(10x^2 + 125)}{(3x^2 - 20)} \] Now, we can't simplify further without knowing the specific value of \(n\), as the remaining terms in the parentheses do not have a common x term with which to factor out or cancel. The simplified expression is: \[ x \cdot \frac{10x^2 + 125}{3x^2 - 20} \]
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