If \(x, y,\) and \(z\) are positive numbers and \(M=\frac{7x^{6}y^{10}}{z^{6}}\), what happens to \(M\) when \(x\) is halved, \(y\) is doubled, and \(z\) is tripled?

(A) it remains the same

(B) it gets multiplied by \(\frac{2}{3}\)

(C) it gets multiplied by \(\frac{7}{3}\)

(D) it gets multiplied by \(\frac{16}{729}\)

(E) it gets multiplied by \(\frac{65536}{729}\)

If \(k\) is a positive integer, which of the following is equivalent to \(\left(13 \times 2^{-k-2} \right) - \left(6 \times 2^{-k-2} \right)?\)

(A) \(\ \ \frac{7}{4 \cdot 2^{k}}\)

(B) \(\ \ \frac{19}{4 \cdot 2^{k}}\)

(C) \(\ \ \frac{7 \cdot 2^{k}}{4}\)

(D) \(\ \ 28 \cdot 2^{k}\)

(E) \(\ \ 76 \cdot 2^{k}\)

Let \(x\), \(y\), and \(z\) be positive numbers. If \(xyz=3\), which of the following equals \(3 \cdot \frac{x^{3}}{y^{2}} \cdot \frac{z^{5}}{y^{-4}} \cdot \frac{1}{xz^{3}}\)?

(A) \(\ \ 1\)

(B) \(\ \ 3\)

(C) \(\ \ 9\)

(D) \(\ \ 27\)

(E) \(\ \ 3x^{2}z^{2}y^{-2}\)

If \(x\) and \(y\) are positive numbers and \(8^{3x} \cdot (64)^{3y}=4096\), what is \(x+2y\)?

(A) \(\ \ \frac{3}{8}\)

(B) \(\ \ \frac{4}{3}\)

(C) \(\ \ 4\)

(D) \(\ \ 512\)

(E) \(\ \ 4096\)

If \(x\) and \(y\) are positive integers and \(6^{3x} \cdot 6^{3y} = 216\) what is \(6x+6y\)?

(A) \(\ \ 0\)

(B) \(\ \ 1\)

(C) \(\ \ 3\)

(D) \(\ \ 6\)

(E) \(\ \ 216\)

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