Solve (x^32x^213x-24)/(x^2)

Question

Simplify the expression

\frac{26 x ^{6} - 24}{x ^{2}}

Evaluate

\frac{x ^{3} \times 2 x ^{2} \times 13 x - 24}{x ^{2}}

Solution

More Steps

Evaluate

x^{3} \times 2 x^{2} \times 13 x

Multiply the terms with the same base by adding their exponents

x^{3 + 2 + 1} \times 2 \times 13

Add the numbers

x^{6} \times 2 \times 13

Multiply the terms

x^{6} \times 26

Use the commutative property to reorder the terms

26 x^{6}

\frac{26 x ^{6} - 24}{x ^{2}}

Show Solution

Find the excluded values

x = 0

Evaluate

\frac{x ^{3} \times 2 x ^{2} \times 13 x - 24}{x ^{2}}

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

x^{2} = 0

Solution

x = 0

Show Solution

Find the roots

x_{1} = - \frac{6 12 \times 1 3 ^{5}}{13}, x_{2} = \frac{6 12 \times 1 3 ^{5}}{13}

Alternative Form

x_{1} \approx - 0.986748, x_{2} \approx 0.986748

Evaluate

\frac{x ^{3} \times 2 x ^{2} \times 13 x - 24}{x ^{2}}

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

\frac{x ^{3} \times 2 x ^{2} \times 13 x - 24}{x ^{2}} = 0

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

\frac{x ^{3} \times 2 x ^{2} \times 13 x - 24}{x ^{2}} = 0, x \neq = 0

Calculate

\frac{x ^{3} \times 2 x ^{2} \times 13 x - 24}{x ^{2}} = 0

Multiply

More Steps

Multiply the terms

x^{3} \times 2 x^{2} \times 13 x

Multiply the terms with the same base by adding their exponents

x^{3 + 2 + 1} \times 2 \times 13

Add the numbers

x^{6} \times 2 \times 13

Multiply the terms

x^{6} \times 26

Use the commutative property to reorder the terms

26 x^{6}

\frac{26 x ^{6} - 24}{x ^{2}} = 0

Cross multiply

26 x^{6} - 24 = x^{2} \times 0

Simplify the equation

26 x^{6} - 24 = 0

Move the constant to the right side

26 x^{6} = 24

Divide both sides

\frac{26 x ^{6}}{26} = \frac{24}{26}

Divide the numbers

x^{6} = \frac{24}{26}

Cancel out the common factor 2

x^{6} = \frac{12}{13}

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

x = \pm 6 \frac{12}{13}

Simplify the expression

More Steps

Evaluate

6 \frac{12}{13}

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

\frac{6 12}{6 13}

Multiply by the Conjugate

\frac{6 12 \times 6 1 3 ^{5}}{6 13 \times 6 1 3 ^{5}}

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

\frac{6 12 \times 1 3 ^{5}}{6 13 \times 6 1 3 ^{5}}

Multiply the numbers

More Steps

Evaluate

613 \times 6 1 3^{5}

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

6 13 \times 1 3^{5}

Calculate the product

6 1 3^{6}

Reduce the index of the radical and exponent with 6

13

\frac{6 12 \times 1 3 ^{5}}{13}

x = \pm \frac{6 12 \times 1 3 ^{5}}{13}

Separate the equation into 2 possible cases

x = \frac{6 12 \times 1 3 ^{5}}{13} x = - \frac{6 12 \times 1 3 ^{5}}{13}

Check if the solution is in the defined range

x = \frac{6 12 \times 1 3 ^{5}}{13} x = - \frac{6 12 \times 1 3 ^{5}}{13}, x \neq = 0

Find the intersection of the solution and the defined range

x = \frac{6 12 \times 1 3 ^{5}}{13} x = - \frac{6 12 \times 1 3 ^{5}}{13}

Solution

x_{1} = - \frac{6 12 \times 1 3 ^{5}}{13}, x_{2} = \frac{6 12 \times 1 3 ^{5}}{13}

Alternative Form

x_{1} \approx - 0.986748, x_{2} \approx 0.986748

Show Solution