Solve (x^2 6x -7)/|x^4|

Question

Simplify the expression

\frac{6 x ^{3} - 7}{x ^{4}}

Evaluate

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣}

Multiply

More Steps

Multiply the terms

x^{2} \times 6 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 6

Add the numbers

x^{3} \times 6

Use the commutative property to reorder the terms

6 x^{3}

\frac{6 x ^{3} - 7}{∣ x ^{4} ∣}

Solution

\frac{6 x ^{3} - 7}{x ^{4}}

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Find the excluded values

x = 0

Evaluate

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣}

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

x^{4} = 0

When the expression in absolute value bars is not negative, remove the bars

x^{4} = 0

Solution

x = 0

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Rewrite the fraction

- \frac{7}{x ^{4}} + \frac{6}{x}

Evaluate

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣}

Evaluate

\frac{6 x ^{3} - 7}{x ^{4}}

For each factor in the denominator,write a new fraction

\frac{?}{x ^{4}} + \frac{?}{x ^{3}} + \frac{?}{x ^{2}} + \frac{?}{x}

Write the terms in the numerator

\frac{A}{x ^{4}} + \frac{B}{x ^{3}} + \frac{C}{x ^{2}} + \frac{D}{x}

Set the sum of fractions equal to the original fraction

\frac{6 x ^{3} - 7}{x ^{4}} = \frac{A}{x ^{4}} + \frac{B}{x ^{3}} + \frac{C}{x ^{2}} + \frac{D}{x}

Multiply both sides

\frac{6 x ^{3} - 7}{x ^{4}} \times x^{4} = \frac{A}{x ^{4}} \times x^{4} + \frac{B}{x ^{3}} \times x^{4} + \frac{C}{x ^{2}} \times x^{4} + \frac{D}{x} \times x^{4}

Simplify the expression

6 x^{3} - 7 = 1 \times A + x B + x^{2} C + x^{3} D

Any expression multiplied by 1 remains the same

6 x^{3} - 7 = A + x B + x^{2} C + x^{3} D

Group the terms

6 x^{3} - 7 = D x^{3} + C x^{2} + B x + A

Equate the coefficients

⎩ ⎨ ⎧ 6 = D 0 = C 0 = B - 7 = A

Swap the sides

⎩ ⎨ ⎧ D = 6 C = 0 B = 0 A = - 7

Find the intersection

⎩ ⎨ ⎧ A = - 7 B = 0 C = 0 D = 6

Solution

- \frac{7}{x ^{4}} + \frac{6}{x}

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Find the roots

x = \frac{3 252}{6}

Alternative Form

x \approx 1.052727

Evaluate

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣}

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

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣} = 0

Find the domain

More Steps

Evaluate

x^{4} \neq = 0

When the expression in absolute value bars is not negative, remove the bars

x^{4} \neq = 0

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

x \neq = 0

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣} = 0, x \neq = 0

Calculate

\frac{x ^{2} \times 6 x - 7}{∣ x ^{4} ∣} = 0

Multiply

More Steps

Multiply the terms

x^{2} \times 6 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 6

Add the numbers

x^{3} \times 6

Use the commutative property to reorder the terms

6 x^{3}

\frac{6 x ^{3} - 7}{∣ x ^{4} ∣} = 0

When the expression in absolute value bars is not negative, remove the bars

\frac{6 x ^{3} - 7}{x ^{4}} = 0

Cross multiply

6 x^{3} - 7 = x^{4} \times 0

Simplify the equation

6 x^{3} - 7 = 0

Move the constant to the right side

6 x^{3} = 7

Divide both sides

\frac{6 x ^{3}}{6} = \frac{7}{6}

Divide the numbers

x^{3} = \frac{7}{6}

Take the 3 -th root on both sides of the equation

3 x^{3} = 3 \frac{7}{6}

Calculate

x = 3 \frac{7}{6}

Simplify the root

More Steps

Evaluate

3 \frac{7}{6}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{3 7}{3 6}

Multiply by the Conjugate

\frac{3 7 \times 3 6 ^{2}}{3 6 \times 3 6 ^{2}}

Simplify

\frac{3 7 \times 3 36}{3 6 \times 3 6 ^{2}}

Multiply the numbers

More Steps

Evaluate

37 \times 336

The product of roots with the same index is equal to the root of the product

3 7 \times 36

Calculate the product

3252

\frac{3 252}{3 6 \times 3 6 ^{2}}

Multiply the numbers

More Steps

Evaluate

36 \times 3 6^{2}

The product of roots with the same index is equal to the root of the product

3 6 \times 6^{2}

Calculate the product

3 6^{3}

Reduce the index of the radical and exponent with 3

6

\frac{3 252}{6}

x = \frac{3 252}{6}

Check if the solution is in the defined range

x = \frac{3 252}{6}, x \neq = 0

Solution

x = \frac{3 252}{6}

Alternative Form

x \approx 1.052727

Show Solution