Example Question - common terms
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Simplifying Algebraic Expression by Factoring
Certainly! To simplify the given expression, we need to factor out common terms and potentially simplify both the numerator and the denominator. The expression is: (10x^(5n+4) + 125x^(5n+2)) / (3x^(5n+3) - 20x^(5n+1)) Now, let's simplify both parts: The numerator can be factored by x^(5n+2), which is the smallest power of x present in both terms: x^(5n+2)(10x^2 + 125) Similarly, the denominator can be factored by x^(5n+1): x^(5n+1)(3x^2 - 20) After factoring out, we get: x^(5n+2)(10x^2 + 125) / x^(5n+1)(3x^2 - 20) Now, we can reduce the expression by eliminating common factors. Notice that x^(5n+2) in the numerator and x^(5n+1) in the denominator can be simplified: x^(5n+2) / x^(5n+1) = x ((5n+2) - (5n+1)) = x^(1) Now we have: x(10x^2 + 125) / (3x^2 - 20) This cannot be further simplified without knowing the specific values of x or unless the quadratic expressions factor into terms that cancel each other out (which is not the case here). Thus the simplified expression is: x(10x^2 + 125) / (3x^2 - 20)
Simplified Expression with Factoring
To simplify the given expression, we will factor common terms and then reduce wherever possible. The expression is: (10 * 5^(n+4) + 125 * 5^(n+2)) / (3 * 5^(n+3) - 20 * 5^(n+1)) Let's factor out the common 5 raised to the smallest power in both the numerator and the denominator. For the numerator, the smallest power of 5 is n+2. We can then express each term with 5^(n+2) factored out: 5^(n+2) * (10 * 5^2 + 125) For the denominator, the smallest power of 5 is n+1. Factor out 5^(n+1): 5^(n+1) * (3 * 5^2 - 20) Now, we can substitute the powers of 5 with its actual value: Numerator becomes: 5^(n+2) * (10 * 25 + 125) 5^(n+2) * (250 + 125) 5^(n+2) * 375 Denominator becomes: 5^(n+1) * (3 * 25 - 20) 5^(n+1) * (75 - 20) 5^(n+1) * 55 Now, put the numerator and denominator back together: (5^(n+2) * 375) / (5^(n+1) * 55) Simplify by cancelling out the common factor of 5: 5^1 in the numerator remains after cancelling out with 5^(n+1) in the denominator and then also simplify the numbers: 5 * 375 / 55 Now, simplifying the numerical part, 375/55 can be reduced: 5 * 375 / 55 = 5 * 75 / 11 The final simplified expression is: (5 * 75) / 11 or 375 / 11
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