Section ASimplify the following algebraic fractions.1) x (x+3)x(x+5)(x-5)7)(x+5)2) y (y-8)9y (2y-1)(y-1)8)y3) x (x+5)9)(x+5)27y² (y-1)x(x+1)(x-1)(x+1)(x-1)(x+1)(x-1)8 (y-16)4)10)8y (y+4)2412y² (y+4)5)5(x-7)11)x (3x-2)10x (x - 7)7x³ (3x-2)²6)3x (3x-4)12)3x3 (5y-3)(y+3)18x² (3x-4)18x4 (5y-3)³
Answer (1)
Answer:It seems you're working on simplifying algebraic fractions. I'm happy to walk you through the process! Let's tackle each problem step-by-step. Remember, the key is to factor the numerator and denominator as much as possible and then cancel out common factors. 1) x(x+3) / x(x+5)(x-5) - Factoring: The numerator and denominator are already factored.- Canceling: We can cancel out the common factor x:plaintext
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x(x+3) / x(x+5)(x-5) = (x+3) / (x+5)(x-5) - Simplified: The simplified fraction is (x+3) / (x+5)(x-5). 2) y(y-8) / 9y(2y-1)(y-1) - Factoring: The numerator and denominator are already factored.- Canceling: We can cancel out the common factor y:plaintext
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y(y-8) / 9y(2y-1)(y-1) = (y-8) / 9(2y-1)(y-1) - Simplified: The simplified fraction is (y-8) / 9(2y-1)(y-1). 3) x(x+5) / 27y² (y-1) - Factoring: The numerator and denominator are already factored.- Canceling: There are no common factors to cancel.- Simplified: The fraction is already in its simplest form: x(x+5) / 27y² (y-1) 4) 8(y-16) / 8y(y+4)² - Factoring: The denominator is already factored; we can factor out 8 from the numerator:plaintext
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8(y-16) / 8y(y+4)² = 8(y-16) / 8y(y+4)(y+4) - Canceling: We can cancel out the common factor 8:plaintext
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8(y-16) / 8y(y+4)² = (y-16) / y(y+4)(y+4) - Simplified: The simplified fraction is (y-16) / y(y+4)². 5) 5(x-7) / 10x(x-7) - Factoring: The numerator and denominator are already factored.- Canceling: We can cancel out the common factors 5 and (x-7):plaintext
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5(x-7) / 10x(x-7) = (1) / 2x - Simplified: The simplified fraction is 1 / 2x. 6) 3x(3x-4) / 18x²(3x-4) - Factoring: The numerator and denominator are already factored.- Canceling: We can cancel out the common factors 3x and (3x-4):plaintext
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3x(3x-4) / 18x²(3x-4) = (1) / 6x - Simplified: The simplified fraction is 1 / 6x. 7) (x+5) / x(x+1)(x-1)(x+1)(x-1) - Factoring: The denominator is already factored. Notice that the factor (x+1) and (x-1) appear twice.- Canceling: We can cancel out the common factor (x+1)(x-1):plaintext
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(x+5) / x(x+1)(x-1)(x+1)(x-1) = (x+5) / x(x+1)(x-1) - Simplified: The simplified fraction is (x+5) / x(x+1)(x-1). 8) y / 12y²(y+4) - Factoring: The denominator is already factored.- Canceling: We can cancel out the common factor y:plaintext
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y / 12y²(y+4) = (1) / 12y(y+4) - Simplified: The simplified fraction is 1 / 12y(y+4). 9) (x+5) / (x+1)(x-1) - Factoring: The numerator and denominator are already factored.- Canceling: There are no common factors to cancel.- Simplified: The fraction is already in its simplest form: (x+5) / (x+1)(x-1). 10) 4 / 8y(y+4)² - Factoring: The denominator is already factored.- Canceling: We can cancel out the common factor 4:plaintext
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4 / 8y(y+4)² = (1) / 2y(y+4)² - Simplified: The simplified fraction is 1 / 2y(y+4)². 11) 5(x-7) / 10x(x - 7) - Factoring: The numerator and denominator are already factored.- Canceling: We can cancel out the common factors 5 and (x-7):plaintext
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5(x-7) / 10x(x-7) = (1) / 2x - Simplified: The simplified fraction is 1 / 2x. 12) 3x³ (5y-3)(y+3) / 18x⁴ (5y-3)³ - Factoring: The numerator and denominator are already factored.- Canceling: We can cancel out the common factors 3x³ and (5y-3)²:plaintext
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3x³ (5y-3)(y+3) / 18x⁴ (5y-3)³ = (y+3) / 6x(5y-3)² - Simplified: The simplified fraction is (y+3) / 6x(5y-3)². Important Math Concepts: - Factoring: Expressing a polynomial as a product of simpler expressions.- Canceling: Removing common factors from the numerator and denominator of a fraction.- Simplifying Fractions: Finding the simplest form of a fraction by reducing it to its lowest terms. Let me know if you have any questions about these steps or would like to work through any other examples! I'm here to help you master the art of simplifying algebraic fractions.Step-by-step explanation:pa brainlies po
