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Five significant digits. Remember that all non-zero digits are significant, and all zeros in-between significant digits are significant.
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Four - zeros between significant digits are significant.
0.007 has one significant figure. The 7 is significant because all non-zero numbers are always significant. The leading zeroes are not significant.
Only one of them is significant. The leading zero (zero to the left) is not significant. The trailing zero (rightmost) is significant. To recap, only the zero following the seven is significant. See the related links for a really good article on significant digits.