Solve (a a^2 a^3)(a-a^2 a^3)

Question

Simplify the expression

a^{7} - a^{11}

Evaluate

(a \times a^{2} \times a^{3}) (a - a^{2} \times a^{3})

Remove the parentheses

a \times a^{2} \times a^{3} (a - a^{2} \times a^{3})

Multiply the terms

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Evaluate

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

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

a^{2 + 3}

Add the numbers

a^{5}

a \times a^{2} \times a^{3} (a - a^{5})

Multiply the terms with the same base by adding their exponents

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

Add the numbers

a^{6} (a - a^{5})

Apply the distributive property

a^{6} \times a - a^{6} \times a^{5}

Multiply the terms

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Evaluate

a^{6} \times a

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

a^{6 + 1}

Add the numbers

a^{7}

a^{7} - a^{6} \times a^{5}

Solution

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Evaluate

a^{6} \times a^{5}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

a^{6 + 5}

Add the numbers

a^{11}

a^{7} - a^{11}

Show Solution

Factor the expression

a^{7} (1 - a) (1 + a) (1 + a^{2})

Evaluate

(a \times a^{2} \times a^{3}) (a - a^{2} \times a^{3})

Remove the parentheses

a \times a^{2} \times a^{3} (a - a^{2} \times a^{3})

Multiply the terms

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Evaluate

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

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

a^{2 + 3}

Add the numbers

a^{5}

a \times a^{2} \times a^{3} (a - a^{5})

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

a^{1 + 2 + 3}

Add the numbers

a^{6}

a^{6} (a - a^{5})

Factor the expression

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Evaluate

a - a^{5}

Rewrite the expression

a - a \times a^{4}

Factor out a from the expression

a (1 - a^{4})

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

a (1 - a^{2}) (1 + a^{2})

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

a (1 - a) (1 + a) (1 + a^{2})

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

Solution

a^{7} (1 - a) (1 + a) (1 + a^{2})

Show Solution

Find the roots

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

Evaluate

(a \times a^{2} \times a^{3}) (a - a^{2} \times a^{3})

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

(a \times a^{2} \times a^{3}) (a - a^{2} \times a^{3}) = 0

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

a^{1 + 2 + 3}

Add the numbers

a^{6}

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

Multiply the terms

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Evaluate

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

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

a^{2 + 3}

Add the numbers

a^{5}

a^{6} (a - a^{5}) = 0

Separate the equation into 2 possible cases

a^{6} = 0 a - a^{5} = 0

The only way a power can be 0 is when the base equals 0

a = 0 a - a^{5} = 0

Solve the equation

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Evaluate

a - a^{5} = 0

Factor the expression

a (1 - a^{4}) = 0

Separate the equation into 2 possible cases

a = 0 1 - a^{4} = 0

Solve the equation

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Evaluate

1 - a^{4} = 0

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

- a^{4} = 0 - 1

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

- a^{4} = - 1

Change the signs on both sides of the equation

a^{4} = 1

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

a = \pm 41

Simplify the expression

a = \pm 1

Separate the equation into 2 possible cases

a = 1 a = - 1

a = 0 a = 1 a = - 1

a = 0 a = 0 a = 1 a = - 1

Find the union

a = 0 a = 1 a = - 1

Solution

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

Show Solution