Notes: `(a + 5)^5` Use the Binomial Theorem to expand and simplify the expression.

Wednesday, July 22, 2009

$\displaystyle{{\left({a}+{5}\right)}}^{{5}}$ Use the Binomial Theorem to expand and simplify the expression.

The above question will be answered with the aid of pascal's triangle. The expansion of the above question using pascal's triangle is expanded as follows:

$\displaystyle{{\left({a}+{b}\right)}}^{{5}}={1}{{a}}^{{5}}+{5}{{a}}^{{4}}{{b}}^{{1}}+{10}{{a}}^{{3}}{{b}}^{{2}}+{10}{{a}}^{{2}}{{b}}^{{3}}+{5}{a}{{b}}^{{4}}+{1}{{b}}^{{5}}$

Now we will use the above equation to solve our problem:

$\displaystyle{{\left({a}+{5}\right)}}^{{5}}={1}\cdot{{a}}^{{5}}+{5}\cdot{a}{4}\cdot{{5}}^{{1}}+{10}\cdot{{a}}^{{3}}\cdot{{5}}^{{2}}+{10}\cdot{{a}}^{{2}}\cdot{{5}}^{{3}}+{5}\cdot{a}\cdot{{\left({5}\right)}}^{{4}}+{{5}}^{{5}}$

Now we can evaluate above, to determine our final answer:

$\displaystyle{{\left({a}+{5}\right)}}^{{5}}={{a}}^{{5}}+{5}\cdot{5}\cdot{{a}}^{{4}}+{10}\cdot{25}\cdot{{a}}^{{3}}+{10}\cdot{125}{{a}}^{{2}}+{5}\cdot{625}\cdot{a}+{3125}$

$\displaystyle{{\left({a}+{5}\right)}}^{{5}}={{a}}^{{5}}+{25}\cdot{{a}}^{{4}}+{250}\cdot{{a}}^{{3}}+{1250}\cdot{{a}}^{{2}}+{3125}\cdot{a}+{3125}$

This is the answer.

Notes

Wednesday, July 22, 2009

$\displaystyle{{\left({a}+{5}\right)}}^{{5}}$ Use the Binomial Theorem to expand and simplify the expression.

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