What is expand (m-n)^6-8m^3 ?

The solution to expand (m-n)^6-8m^3 is m^6-6m^5n+15m^4n^2-20m^3n^3+15m^2n^4-6mn^5+n^6-8m^3

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expand (m-n)^6-8m^3

expand

(m - n)^{6} - 8 m^{3}

m^{6} - 6 m^{5} n + 15 m^{4} n^{2} - 20 m^{3} n^{3} + 15 m^{2} n^{4} - 6 m n^{5} + n^{6} - 8 m^{3}

Solution steps

(m - n)^{6} - 8 m^{3}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = m, b = - n

= i = 0 \sum 6 (i 6) m^{(6 - i)} (- n)^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} m^{6} (- n)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} m^{5} (- n)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} m^{4} (- n)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} m^{3} (- n)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} m^{2} (- n)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} m^{1} (- n)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} m^{0} (- n)^{6} - 8 m^{3}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} m^{6} (- n)^{0} : m^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} m^{5} (- n)^{1} : - 6 m^{5} n

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} m^{4} (- n)^{2} : 15 m^{4} n^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} m^{3} (- n)^{3} : - 20 m^{3} n^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} m^{2} (- n)^{4} : 15 m^{2} n^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} m^{1} (- n)^{5} : - 6 m n^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} m^{0} (- n)^{6} : n^{6}

= m^{6} - 6 m^{5} n + 15 m^{4} n^{2} - 20 m^{3} n^{3} + 15 m^{2} n^{4} - 6 m n^{5} + n^{6} - 8 m^{3}

\frac{1}{cos ^{2} ( x )} - \frac{1}{sin ^{2} ( x )}

13 a - 13 b

\frac{- ( 32 a ^{2} \cdot b )}{( 4 a \cdot b ^{5} )}

3 (3 x + 2 y)

2 n^{2} + 4 n - 1260

What is expand (m-n)^6-8m^3 ?
The solution to expand (m-n)^6-8m^3 is m^6-6m^5n+15m^4n^2-20m^3n^3+15m^2n^4-6mn^5+n^6-8m^3