(2k-5)/(32k^(2)+8k)*(5k+2)/(7)

Factor Denominator: First, we need to factor the denominator 32k^2 + 8k to see if there are any common factors with the numerator 2k - 5. Factoring out 8k from the denominator gives us 8k(4k + 1).Check Common Factors: Now, we look for common factors between the numerators (2k-5) and (5k+2) and the factored denominator 8k(4k + 1). There are no obvious common factors.Multiply Fractions: Next, we multiply the two fractions together. When multiplying fractions, we multiply the numerators together and the denominators together. So, (2k-5)*(5k+2) becomes the new numerator and 8k(4k + 1)*7 becomes the new denominator.Multiply Numerator: We perform the multiplication for the numerator (2k-5)*(5k+2): This gives us 10k^2 + 4k - 25k - 10 which simplifies to 10k^2 - 21k - 10.Multiply Denominator: We perform the multiplication for the denominator 8k(4k + 1)*7: This gives us 8k*4k*7 + 8k*1*7 which simplifies to 224k^2 + 56k.Write Simplified Expression: Now we write down the simplified form of the expression with the new numerator and denominator: The expression is (10k^2 - 21k - 10)/(224k^2 + 56k).Check Further Simplification: Finally, we check if the expression can be simplified further by factoring, but since there are no common factors between the numerator and the denominator, this is the simplified form of the expression.

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$\frac{2 k-5}{32 k^{2}+8 k} \cdot \frac{5 k+2}{7}$

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Algebra 2

Sum of finite series starts from 1

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\frac{2 k-5}{32 k^{2}+8 k} \cdot \frac{5 k+2}{7}

Factor Denominator: First, we need to factor the denominator $32k^2 + 8k$ to see if there are any common factors with the numerator $2k - 5$ . Factoring out $8k$ from the denominator gives us $8k(4k + 1)$ .
Check Common Factors: Now, we look for common factors between the numerators $2k-5$ and $5k+2$ and the factored denominator $8k(4k + 1)$ . There are no obvious common factors.
Multiply Fractions: Next, we multiply the two fractions together. When multiplying fractions, we multiply the numerators together and the denominators together. $\newline$ So, $(2k-5)\times(5k+2)$ becomes the new numerator and $8k(4k + 1)\times7$ becomes the new denominator.
Multiply Numerator: We perform the multiplication for the numerator $(2k-5)\cdot(5k+2)$ : This gives us $10k^2 + 4k - 25k - 10$ which simplifies to $10k^2 - 21k - 10$ .
Multiply Denominator: We perform the multiplication for the denominator $8k(4k + 1)*7$ : This gives us $8k*4k*7 + 8k*1*7$ which simplifies to $224k^2 + 56k$ .
Write Simplified Expression: Now we write down the simplified form of the expression with the new numerator and denominator: $\newline$ The expression is $(10k^2 - 21k - 10)/(224k^2 + 56k)$ .
Check Further Simplification: Finally, we check if the expression can be simplified further by factoring, but since there are no common factors between the numerator and the denominator, this is the simplified form of the expression.