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Simplify: $-\frac{1}{2}a^{2}b^{2}c+\frac{1}{3}ab^{2}c-\frac{1}{4}abc^{2}-\frac{1}{5}cb^{2}a^{2}+\frac{1}{6}cb^{2}a-\frac{1}{7}c^{2}ab+\frac{1}{8}ca^{2}b$.
Given:
$-\frac{1}{2}a^{2}b^{2}c+\frac{1}{3}ab^{2}c-\frac{1}{4}abc^{2}-\frac{1}{5}cb^{2}a^{2}+\frac{1}{6}cb^{2}a-\frac{1}{7}c^{2}ab+\frac{1}{8}ca^{2}b$.
To do:
We have to simplify the given expression.
Solution:
Taking common terms together and simplifying, we get,
$-\frac{1}{2}a^{2}b^{2}c+\frac{1}{3}ab^{2}c-\frac{1}{4}abc^{2}-\frac{1}{5}cb^{2}a^{2}+\frac{1}{6}cb^{2}a-\frac{1}{7}c^{2}ab+\frac{1}{8}ca^{2}b=a^{2}b^{2}c(-\frac{1}{2}-\frac{1}{5})+ab^{2}c(\frac{1}{3}+\frac{1}{6})+abc^{2}(-\frac{1}{4}-\frac{1}{7})+\frac{1}{8}ca^{2}b$
$=a^{2}b^{2}c[-(\frac{1(5)+1(2)}{10})]+ab^{2}c(\frac{1(2)+1}{6})+abc^{2}[-(\frac{1(7)+1(4)}{28})]+\frac{1}{8}ca^{2}b$
$=a^{2}b^{2}c[-(\frac{5+2}{10})]+ab^{2}c(\frac{2+1}{6})+abc^{2}[-(\frac{7+4}{28})]+\frac{1}{8}ca^{2}b$
$=a^{2}b^{2}c[-(\frac{7}{10})]+ab^{2}c(\frac{3}{6})+abc^{2}[-(\frac{11}{28})]+\frac{1}{8}ca^{2}b$
$=-\frac{7}{10}a^{2}b^{2}c+\frac{1}{2}ab^{2}c-\frac{11}{28}abc^{2}+\frac{1}{8}ca^{2}b$
Therefore,
$-\frac{1}{2}a^{2}b^{2}c+\frac{1}{3}ab^{2}c-\frac{1}{4}abc^{2}-\frac{1}{5}cb^{2}a^{2}+\frac{1}{6}cb^{2}a-\frac{1}{7}c^{2}ab+\frac{1}{8}ca^{2}b=-\frac{7}{10}a^{2}b^{2}c+\frac{1}{2}ab^{2}c-\frac{11}{28}abc^{2}+\frac{1}{8}ca^{2}b$.
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