Prove the following using a proof by contrapositive:. For every pair of real numbers x and y, if x is rational and xy is irrational, then y is irrational.
Let q = a/b and r = c/b be two rational numbers written in lowest terms. Let s = q+r and s = e. f written in lowest terms. Assume that s is not 0.
Prove or disprove the following two statements.
a. If b and d are odd, then f is odd.
b. If b and d are even, then f is even.

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If the ratio of sum of the first m and n terms of an ap is m2 : n2 , show that the ratio of its mth and nth terms is (2m − 1) : (2n − 1).

The edge of a cube was found to be 30 cm with a possible error in measurement of 0.4 cm. use differentials to estimate the maximum possible error, relative error, and percentage error in computing the volume of the cube and the surface area of the cube. (round your answers to four decimal places.) (a) the volume of the cube maximum possible error cm3 relative error percentage error % (b) the surface area of the cube maximum possible error cm2 relative error percentage error %