(-3*a-3*b-3*c)^2 — step-by-step solution

Let's evaluate:

{(- 3 a - 3 b - 3 c)}^{2}

ⓘ The minus sign is factored out, changing the sign of each term:

- 3 a - 3 b - 3 c

- (3 a + 3 b + 3 c)

■With an even integer exponent, the minus sign in the base disappears; with an odd integer exponent, it moves in front of the power:

{(- (3 a + 3 b + 3 c))}^{2}

{(3 a + 3 b + 3 c)}^{2}

{(3 a + 3 b + 3 c)}^{2}

ⓘ Factor out the common numerical factor

3

before expanding the power:

{(3 a + 3 b + 3 c)}^{2}

3^{2} \cdot {(a + b + c)}^{2}

3^{2} \cdot {(a + b + c)}^{2}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

3^{2}

3 \cdot 3

9

9 \cdot {(a + b + c)}^{2}

Using the multinomial theorem:

{(x 1 + x 2 + \dots + x k)}^{n} = \sum_{m 1 + m 2 + \dots + m k = n}^{m i \geq 0} (\frac{n!}{m 1! m 2! \dots m k!}) {x 1}^{m 1} {x 2}^{m 2} \dots {x k}^{m k}

, you can expand the math expression:

ⅰ

{(a + b + c)}^{2}

\frac{2!}{2! \cdot 0! \cdot 0!} \cdot a^{2} b^{0} c^{0}

+ \frac{2!}{1! \cdot 1! \cdot 0!} \cdot a^{1} b^{1} c^{0}

+ \frac{2!}{0! \cdot 2! \cdot 0!} \cdot a^{0} b^{2} c^{0}

+ \frac{2!}{1! \cdot 0! \cdot 1!} \cdot a^{1} b^{0} c^{1}

+ \frac{2!}{0! \cdot 1! \cdot 1!} \cdot a^{0} b^{1} c^{1}

+ \frac{2!}{0! \cdot 0! \cdot 2!} \cdot a^{0} b^{0} c^{2}

9 \cdot (\frac{2!}{2! \cdot 0! \cdot 0!} \cdot a^{2} b^{0} c^{0}

+ \frac{2!}{1! \cdot 1! \cdot 0!} \cdot a^{1} b^{1} c^{0}

+ \frac{2!}{0! \cdot 2! \cdot 0!} \cdot a^{0} b^{2} c^{0}

+ \frac{2!}{1! \cdot 0! \cdot 1!} \cdot a^{1} b^{0} c^{1}

+ \frac{2!}{0! \cdot 1! \cdot 1!} \cdot a^{0} b^{1} c^{1}

+ \frac{2!}{0! \cdot 0! \cdot 2!} \cdot a^{0} b^{0} c^{2})

ⓘ Any nonzero number raised to the power 0 is 1. In this calculation, we use the convention 0^0 = 1:

b^{0}

1

c^{0}

1

a^{0}

1

9 \cdot (\frac{2!}{2! \cdot 0! \cdot 0!} \cdot a^{2}

+ \frac{2!}{1! \cdot 1! \cdot 0!} \cdot a b

+ \frac{2!}{0! \cdot 2! \cdot 0!} \cdot b^{2}

+ \frac{2!}{1! \cdot 0! \cdot 1!} \cdot a c

+ \frac{2!}{0! \cdot 1! \cdot 1!} \cdot b c

+ \frac{2!}{0! \cdot 0! \cdot 2!} \cdot c^{2})

ⓘ The factorial of a number is the product of all the integers from 1 to that number:

2!

2

1!

1

■By convention, the factorial of 0 is 1:

0!

1

■Multiplying by 1 changes nothing:

2 \cdot 1 \cdot 1

2

1 \cdot 1 \cdot 1

1

1 \cdot 2 \cdot 1

2

1 \cdot 1 \cdot 2

2

9 \cdot (\frac{2}{2} \cdot a^{2}

+ \frac{2}{1} \cdot a b

+ \frac{2}{2} \cdot b^{2}

+ \frac{2}{1} \cdot a c

+ \frac{2}{1} \cdot b c

+ \frac{2}{2} \cdot c^{2})

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{2}{2}

1

■Multiplying by 1 changes nothing:

1 \cdot a^{2}

a^{2}

1 \cdot b^{2}

b^{2}

1 \cdot c^{2}

c^{2}

9 \cdot (a^{2}

+ 2 a b

+ b^{2}

+ 2 a c

+ 2 b c

+ c^{2})

ⓘ Expand by multiplying each term inside the parentheses:

9 \cdot (a^{2}

+ 2 a b

+ b^{2}

+ 2 a c

+ 2 b c

+ c^{2})

9 a^{2}

+ 18 a b

+ 9 b^{2}

+ 18 a c

+ 18 b c

+ 9 c^{2}

9 a^{2}

+ 18 a b

+ 9 b^{2}

+ 18 a c

+ 18 b c

+ 9 c^{2}

▶Result:

9 a^{2}

+ 18 a b

+ 18 a c

+ 9 b^{2}

+ 18 b c

+ 9 c^{2}