Prove this theorem. Use prime factorization to prove the "Fact"

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Suppose z varies directly with y and directly with the cube of x. If z x = 5 and y = 7, what is z when x = 6 and y = 3? = z (Round your answer to one decima place) 2012.5 when

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Suppose that y is inversely proportional to square root of x. Find the constant of proportionality k if y = 8 when x = 10. k (Round answer to 3 decimal places) Using the k from above write the variation equation in terms of x. y Find the value of y when x У = 34. (Round answer to 3 decimal places) Enter DNE if answer does not exist.

y varies inversely as x. If x = 2 then y = 15. Find y when x = 5. y =

y varies directly as x. If x = -4 then y = 10. Find y when x = 2. Write answer as reduced fraction if answer is not integer. y =

Prove that for all positive integers n, the number (27) is even. (Hint: Use Pascal's Identity.)

Using the Binomial Theorem, find the coefficient x4 in the expansion of x(1 − x)² + x²(1 + 2x)³ + x³ (1 + 3x) 12.