Question - Simplifying Expression with Exponents
mathquadratic expressionssimplifying expressions with exponentslaws of exponentsfactor out common terms
Solution:
To simplify the expression given in the image, we need to follow the laws of exponents for division and addition. The expression is:\[ \frac{10x^{n+4} + 4.125x^{n+2}}{3x^{n+3} - 20x^{n+1}} \]First, factor out the common terms involving $$x$$ from the numerator and the denominator.1. In the numerator, factor out the smallest power of $$x$$ which is $$x^{n+2}$$:\[ x^{n+2}(10x^{2} + 4.125) \]2. In the denominator, factor out the smallest power of $$x$$ which is $$x^{n+1}$$:\[ x^{n+1}(3x^{2} - 20) \]Now we rewrite the expression:\[ \frac{x^{n+2}(10x^{2} + 4.125)}{x^{n+1}(3x^{2} - 20)} \]Next, cancel out the common $$x^{n+1}$$ term:\[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \]Since $$x$$ does not cancel anything further in the denominator, we leave the expression as is.The final simplified expression is:\[ \frac{x(10x^{2} + 4.125)}{3x^{2} - 20} \]Depending on the context or further instructions, there might be additional simplification possible, like factoring the quadratic expressions if they are factorable. However, without additional context, this is the simplified form based on the given expression.
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