Solve (27n-25)/(-9n^(2) 27n-20) ⋅ (30n^(3)-40n^(2))/(

Question

Simplify the expression

- \frac{810 n ^{4} - 1830 n ^{3} + 1000 n ^{2}}{2187 n ^{4} - 17496 n ^{3} + 180 n - 1440}

Evaluate

\frac{27 n - 25}{- 9 n ^{2} \times 27 n - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72}

Multiply

More Steps

\frac{27 n - 25}{- 243 n ^{3} - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{27 n - 25}{243 n ^{3} + 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72}

Multiply the terms

- \frac{( 27 n - 25 ) ( 30 n ^{3} - 40 n ^{2} )}{( 243 n ^{3} + 20 ) ( 9 n - 72 )}

Multiply the terms

More Steps

- \frac{810 n ^{4} - 1830 n ^{3} + 1000 n ^{2}}{( 243 n ^{3} + 20 ) ( 9 n - 72 )}

Solution

More Steps

- \frac{810 n ^{4} - 1830 n ^{3} + 1000 n ^{2}}{2187 n ^{4} - 17496 n ^{3} + 180 n - 1440}

Show Solution

n = - \frac{3 60}{9}, n = 8

Evaluate

\frac{27 n - 25}{- 9 n ^{2} \times 27 n - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72}

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

- 9 n^{2} \times 27 n - 20 = 0 9 n - 72 = 0

Solve the equations

More Steps

n = - \frac{3 60}{9} 9 n - 72 = 0

Solve the equations

More Steps

n = - \frac{3 60}{9} n = 8

Solution

n = - \frac{3 60}{9}, n = 8

Show Solution

n_{1} = 0, n_{2} = \frac{25}{27}, n_{3} = \frac{4}{3}

Alternative Form

n_{1} = 0, n_{2} = 0. \dot{9} 2 \dot{5}, n_{3} = 1. \dot{3}

Evaluate

\frac{27 n - 25}{- 9 n ^{2} \times 27 n - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72}

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

\frac{27 n - 25}{- 9 n ^{2} \times 27 n - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72} = 0

Find the domain

More Steps

\frac{27 n - 25}{- 9 n ^{2} \times 27 n - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72} = 0, n \in (- \infty, - \frac{3 60}{9}) \cup (- \frac{3 60}{9}, 8) \cup (8, + \infty)

Calculate

\frac{27 n - 25}{- 9 n ^{2} \times 27 n - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72} = 0

Multiply

More Steps

\frac{27 n - 25}{- 243 n ^{3} - 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72} = 0

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{27 n - 25}{243 n ^{3} + 20} \times \frac{30 n ^{3} - 40 n ^{2}}{9 n - 72} = 0

Multiply the terms

- \frac{( 27 n - 25 ) ( 30 n ^{3} - 40 n ^{2} )}{( 243 n ^{3} + 20 ) ( 9 n - 72 )} = 0

Rewrite the expression

\frac{( - 27 n + 25 ) ( 30 n ^{3} - 40 n ^{2} )}{( 243 n ^{3} + 20 ) ( 9 n - 72 )} = 0

Cross multiply

(- 27 n + 25) (30 n^{3} - 40 n^{2}) = (243 n^{3} + 20) (9 n - 72) \times 0

Simplify the equation

(- 27 n + 25) (30 n^{3} - 40 n^{2}) = 0

Change the sign

(27 n - 25) (30 n^{3} - 40 n^{2}) = 0

Separate the equation into 2 possible cases

27 n - 25 = 0 30 n^{3} - 40 n^{2} = 0

Solve the equation

More Steps

n = \frac{25}{27} 30 n^{3} - 40 n^{2} = 0

Solve the equation

More Steps

n = \frac{25}{27} n = 0 n = \frac{4}{3}

Check if the solution is in the defined range

n = \frac{25}{27} n = 0 n = \frac{4}{3}, n \in (- \infty, - \frac{3 60}{9}) \cup (- \frac{3 60}{9}, 8) \cup (8, + \infty)

Find the intersection of the solution and the defined range

n = \frac{25}{27} n = 0 n = \frac{4}{3}

Solution

n_{1} = 0, n_{2} = \frac{25}{27}, n_{3} = \frac{4}{3}

Alternative Form

n_{1} = 0, n_{2} = 0. \dot{9} 2 \dot{5}, n_{3} = 1. \dot{3}

Show Solution