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(B) Let the two geometric means between $1$ and $64$ be $a$ and $b$.Then the sequence $1, a, b, 64$ forms a Geometric Progression $(GP)$.Let the commo

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$4$ and $16$

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(B) Let the two geometric means between $1$ and $64$ be $a$ and $b$.Then the sequence $1, a, b, 64$ forms a Geometric Progression $(GP)$.Let the common ratio be $r$.Then $a = 1 \cdot r$,$b = 1 \cdot r^2$,and $64 = 1 \cdot r^3$.From $r^3 = 64$,we get $r = \sqrt[3]{64} = 4$.Therefore,$a = 1 \cdot 4 = 4$ and $b = 1 \cdot 4^2 = 16$.Thus,the two geometric means are $4$ and $16$.

The two geometric means between the numbers $1$ and $64$ are

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