Solve a^3 a^3 6a^2 6a^6a-36a

Question

Simplify the expression

36 a^{15} - 36 a

Evaluate

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a - 36 a

Solution

More Steps

Evaluate

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a

Multiply the terms with the same base by adding their exponents

a^{3 + 3 + 2 + 6 + 1} \times 6 \times 6

Add the numbers

a^{15} \times 6 \times 6

Multiply the terms

a^{15} \times 36

Use the commutative property to reorder the terms

36 a^{15}

36 a^{15} - 36 a

Show Solution

Factor the expression

36 a (a - 1) (a + 1) (a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1) (a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1)

Evaluate

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a - 36 a

Multiply

More Steps

Evaluate

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a

Multiply the terms with the same base by adding their exponents

a^{3 + 3 + 2 + 6 + 1} \times 6 \times 6

Add the numbers

a^{15} \times 6 \times 6

Multiply the terms

a^{15} \times 36

Use the commutative property to reorder the terms

36 a^{15}

36 a^{15} - 36 a

Rewrite the expression

36 a \times a^{14} - 36 a

Factor out 36 a from the expression

36 a (a^{14} - 1)

Factor the expression

More Steps

Evaluate

a^{14} - 1

Calculate

a^{14} + a^{13} - a^{7} - a^{6} - a^{13} - a^{12} + a^{6} + a^{5} + a^{12} + a^{11} - a^{5} - a^{4} - a^{11} - a^{10} + a^{4} + a^{3} + a^{10} + a^{9} - a^{3} - a^{2} - a^{9} - a^{8} + a^{2} + a + a^{8} + a^{7} - a - 1

Rewrite the expression

a^{6} \times a^{8} + a^{6} \times a^{7} - a^{6} \times a - a^{6} - a^{5} \times a^{8} - a^{5} \times a^{7} + a^{5} \times a + a^{5} + a^{4} \times a^{8} + a^{4} \times a^{7} - a^{4} \times a - a^{4} - a^{3} \times a^{8} - a^{3} \times a^{7} + a^{3} \times a + a^{3} + a^{2} \times a^{8} + a^{2} \times a^{7} - a^{2} \times a - a^{2} - a \times a^{8} - a \times a^{7} + a \times a + a + a^{8} + a^{7} - a - 1

Factor out a^{6} from the expression

a^{6} (a^{8} + a^{7} - a - 1) - a^{5} \times a^{8} - a^{5} \times a^{7} + a^{5} \times a + a^{5} + a^{4} \times a^{8} + a^{4} \times a^{7} - a^{4} \times a - a^{4} - a^{3} \times a^{8} - a^{3} \times a^{7} + a^{3} \times a + a^{3} + a^{2} \times a^{8} + a^{2} \times a^{7} - a^{2} \times a - a^{2} - a \times a^{8} - a \times a^{7} + a \times a + a + a^{8} + a^{7} - a - 1

Factor out - a^{5} from the expression

a^{6} (a^{8} + a^{7} - a - 1) - a^{5} (a^{8} + a^{7} - a - 1) + a^{4} \times a^{8} + a^{4} \times a^{7} - a^{4} \times a - a^{4} - a^{3} \times a^{8} - a^{3} \times a^{7} + a^{3} \times a + a^{3} + a^{2} \times a^{8} + a^{2} \times a^{7} - a^{2} \times a - a^{2} - a \times a^{8} - a \times a^{7} + a \times a + a + a^{8} + a^{7} - a - 1

Factor out a^{4} from the expression

a^{6} (a^{8} + a^{7} - a - 1) - a^{5} (a^{8} + a^{7} - a - 1) + a^{4} (a^{8} + a^{7} - a - 1) - a^{3} \times a^{8} - a^{3} \times a^{7} + a^{3} \times a + a^{3} + a^{2} \times a^{8} + a^{2} \times a^{7} - a^{2} \times a - a^{2} - a \times a^{8} - a \times a^{7} + a \times a + a + a^{8} + a^{7} - a - 1

Factor out - a^{3} from the expression

a^{6} (a^{8} + a^{7} - a - 1) - a^{5} (a^{8} + a^{7} - a - 1) + a^{4} (a^{8} + a^{7} - a - 1) - a^{3} (a^{8} + a^{7} - a - 1) + a^{2} \times a^{8} + a^{2} \times a^{7} - a^{2} \times a - a^{2} - a \times a^{8} - a \times a^{7} + a \times a + a + a^{8} + a^{7} - a - 1

Factor out a^{2} from the expression

a^{6} (a^{8} + a^{7} - a - 1) - a^{5} (a^{8} + a^{7} - a - 1) + a^{4} (a^{8} + a^{7} - a - 1) - a^{3} (a^{8} + a^{7} - a - 1) + a^{2} (a^{8} + a^{7} - a - 1) - a \times a^{8} - a \times a^{7} + a \times a + a + a^{8} + a^{7} - a - 1

Factor out - a from the expression

a^{6} (a^{8} + a^{7} - a - 1) - a^{5} (a^{8} + a^{7} - a - 1) + a^{4} (a^{8} + a^{7} - a - 1) - a^{3} (a^{8} + a^{7} - a - 1) + a^{2} (a^{8} + a^{7} - a - 1) - a (a^{8} + a^{7} - a - 1) + a^{8} + a^{7} - a - 1

Factor out a^{8} + a^{7} - a - 1 from the expression

(a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1) (a^{8} + a^{7} - a - 1)

36 a (a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1) (a^{8} + a^{7} - a - 1)

Factor the expression

More Steps

Evaluate

a^{8} + a^{7} - a - 1

Calculate

a^{8} - a^{6} + a^{7} - a^{5} + a^{6} - a^{4} + a^{5} - a^{3} + a^{4} - a^{2} + a^{3} - a + a^{2} - 1

Rewrite the expression

a^{6} \times a^{2} - a^{6} + a^{5} \times a^{2} - a^{5} + a^{4} \times a^{2} - a^{4} + a^{3} \times a^{2} - a^{3} + a^{2} \times a^{2} - a^{2} + a \times a^{2} - a + a^{2} - 1

Factor out a^{6} from the expression

a^{6} (a^{2} - 1) + a^{5} \times a^{2} - a^{5} + a^{4} \times a^{2} - a^{4} + a^{3} \times a^{2} - a^{3} + a^{2} \times a^{2} - a^{2} + a \times a^{2} - a + a^{2} - 1

Factor out a^{5} from the expression

a^{6} (a^{2} - 1) + a^{5} (a^{2} - 1) + a^{4} \times a^{2} - a^{4} + a^{3} \times a^{2} - a^{3} + a^{2} \times a^{2} - a^{2} + a \times a^{2} - a + a^{2} - 1

Factor out a^{4} from the expression

a^{6} (a^{2} - 1) + a^{5} (a^{2} - 1) + a^{4} (a^{2} - 1) + a^{3} \times a^{2} - a^{3} + a^{2} \times a^{2} - a^{2} + a \times a^{2} - a + a^{2} - 1

Factor out a^{3} from the expression

a^{6} (a^{2} - 1) + a^{5} (a^{2} - 1) + a^{4} (a^{2} - 1) + a^{3} (a^{2} - 1) + a^{2} \times a^{2} - a^{2} + a \times a^{2} - a + a^{2} - 1

Factor out a^{2} from the expression

a^{6} (a^{2} - 1) + a^{5} (a^{2} - 1) + a^{4} (a^{2} - 1) + a^{3} (a^{2} - 1) + a^{2} (a^{2} - 1) + a \times a^{2} - a + a^{2} - 1

Factor out a from the expression

a^{6} (a^{2} - 1) + a^{5} (a^{2} - 1) + a^{4} (a^{2} - 1) + a^{3} (a^{2} - 1) + a^{2} (a^{2} - 1) + a (a^{2} - 1) + a^{2} - 1

Factor out a^{2} - 1 from the expression

(a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) (a^{2} - 1)

36 a (a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1) (a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) (a^{2} - 1)

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

36 a (a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1) (a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1) (a - 1) (a + 1)

Solution

36 a (a - 1) (a + 1) (a^{6} - a^{5} + a^{4} - a^{3} + a^{2} - a + 1) (a^{6} + a^{5} + a^{4} + a^{3} + a^{2} + a + 1)

Show Solution

Find the roots

a_{1} = - 1, a_{2} = 0, a_{3} = 1

Evaluate

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a - 36 a

To find the roots of the expression,set the expression equal to 0

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a - 36 a = 0

Multiply

More Steps

Multiply the terms

a^{3} \times a^{3} \times 6 a^{2} \times 6 a^{6} \times a

Multiply the terms with the same base by adding their exponents

a^{3 + 3 + 2 + 6 + 1} \times 6 \times 6

Add the numbers

a^{15} \times 6 \times 6

Multiply the terms

a^{15} \times 36

Use the commutative property to reorder the terms

36 a^{15}

36 a^{15} - 36 a = 0

Factor the expression

36 a (a^{14} - 1) = 0

Divide both sides

a (a^{14} - 1) = 0

Separate the equation into 2 possible cases

a = 0 a^{14} - 1 = 0

Solve the equation

More Steps

Evaluate

a^{14} - 1 = 0

Move the constant to the right-hand side and change its sign

a^{14} = 0 + 1

Removing 0 doesn't change the value,so remove it from the expression

a^{14} = 1

Take the root of both sides of the equation and remember to use both positive and negative roots

a = \pm 141

Simplify the expression

a = \pm 1

Separate the equation into 2 possible cases

a = 1 a = - 1

a = 0 a = 1 a = - 1

Solution

a_{1} = - 1, a_{2} = 0, a_{3} = 1

Show Solution