Solve (x^(2) 7x 12)/(x^3) / (x-1)/(x^4)

Question

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}}

Simplify the expression

\frac{84 x ^{4}}{x - 1}

Evaluate

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}}

Reduce the fraction

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Evaluate

\frac{x ^{2} \times 7 x \times 12}{x ^{3}}

Multiply

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Evaluate

x^{2} \times 7 x \times 12

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 7 \times 12

Add the numbers

x^{3} \times 7 \times 12

Multiply the terms

x^{3} \times 84

\frac{x ^{3} \times 84}{x ^{3}}

Reduce the fraction

84

84 \div \frac{x - 1}{x ^{4}}

Multiply by the reciprocal

84 \times \frac{x ^{4}}{x - 1}

Solution

\frac{84 x ^{4}}{x - 1}

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Find the excluded values

x = 0, x = 1

Evaluate

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}}

To find the excluded values,set the denominators equal to 0

x^{3} = 0 x^{4} = 0 \frac{x - 1}{x ^{4}} = 0

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

x = 0 x^{4} = 0 \frac{x - 1}{x ^{4}} = 0

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

x = 0 x = 0 \frac{x - 1}{x ^{4}} = 0

Solve the equations

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Evaluate

\frac{x - 1}{x ^{4}} = 0

Cross multiply

x - 1 = x^{4} \times 0

Simplify the equation

x - 1 = 0

Move the constant to the right side

x = 0 + 1

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

x = 1

x = 0 x = 0 x = 1

Solution

x = 0, x = 1

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Find the roots

x \in \emptyset

Evaluate

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}}

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

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}} = 0

Find the domain

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Evaluate

⎩ ⎨ ⎧ x^{3} \neq = 0 x^{4} \neq = 0 \frac{x - 1}{x ^{4}} \neq = 0

The only way a power can not be 0 is when the base not equals 0

⎩ ⎨ ⎧ x \neq = 0 x^{4} \neq = 0 \frac{x - 1}{x ^{4}} \neq = 0

The only way a power can not be 0 is when the base not equals 0

⎩ ⎨ ⎧ x \neq = 0 x \neq = 0 \frac{x - 1}{x ^{4}} \neq = 0

Calculate

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Evaluate

\frac{x - 1}{x ^{4}} \neq = 0

Multiply both sides

\frac{x - 1}{x ^{4}} \times x^{4} \neq = 0 \times x^{4}

Evaluate

x - 1 \neq = 0 \times x^{4}

Multiply both sides

x - 1 \neq = 0

Move the constant to the right side

x \neq = 0 + 1

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

x \neq = 1

⎩ ⎨ ⎧ x \neq = 0 x \neq = 0 x \neq = 1

Simplify

{x \neq = 0 x \neq = 1

Find the intersection

x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}} = 0, x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Calculate

\frac{x ^{2} \times 7 x \times 12}{x ^{3}} \div \frac{x - 1}{x ^{4}} = 0

Multiply

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Multiply the terms

x^{2} \times 7 x \times 12

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 7 \times 12

Add the numbers

x^{3} \times 7 \times 12

Multiply the terms

x^{3} \times 84

Use the commutative property to reorder the terms

84 x^{3}

\frac{84 x ^{3}}{x ^{3}} \div \frac{x - 1}{x ^{4}} = 0

Divide the terms

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Evaluate

\frac{84 x ^{3}}{x ^{3}}

Reduce the fraction

\frac{84}{1}

Divide the terms

84

84 \div \frac{x - 1}{x ^{4}} = 0

Divide the terms

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Evaluate

84 \div \frac{x - 1}{x ^{4}}

Multiply by the reciprocal

84 \times \frac{x ^{4}}{x - 1}

Multiply the terms

\frac{84 x ^{4}}{x - 1}

\frac{84 x ^{4}}{x - 1} = 0

Cross multiply

84 x^{4} = (x - 1) \times 0

Simplify the equation

84 x^{4} = 0

Rewrite the expression

x^{4} = 0

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

x = 0

Check if the solution is in the defined range

x = 0, x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Solution

x \in \emptyset

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