Question - Simplifying Algebraic Expressions with Exponents
Solution:
To simplify the given expression, you'll need to use the laws of exponents. Let's simplify the numerator and the denominator separately first, and then we'll see if we can simplify further.The expression given is:$$ \frac{10x^{5n+4} + 4.125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} $$Numerator:$$ 10x^{5n+4} + 4.125x^{5n+2} $$ can be factored by taking out the common factor of $$ x^{5n+2} $$:$$ = x^{5n+2} (10x^2 + 4.125) $$Denominator:$$ 3x^{5n+3} - 20x^{5n+1} $$ similarly can be factored by taking out the common factor of $$ x^{5n+1} $$:$$ = x^{5n+1} (3x^2 - 20) $$Now the expression becomes:$$ \frac{x^{5n+2} (10x^2 + 4.125)}{x^{5n+1} (3x^2 - 20)} $$Next, we'll divide the exponents:Since $$ x^{5n+2} $$ is divided by $$ x^{5n+1} $$, we subtract the exponents of like bases:$$ x^{5n+2} ÷ x^{5n+1} = x^{(5n+2) - (5n+1)} = x^{1} = x $$Now we have:$$ \frac{x(10x^2 + 4.125)}{3x^2 - 20} $$The expression is simplified to:$$ \frac{10x^3 + 4.125x}{3x^2 - 20} $$This is as simple as the expression can get without further information about $$ x $$ or $$ n $$. If certain values are given for these variables, then numerical simplification could proceed. Otherwise, this is the simplified algebraic expression.
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