Solve (k^(2)-36)/(k^(2)-16k^60) / (k^(2)-12k^36)/(k^(

Question

Simplify the expression

\frac{k ^{2} - 4 k - 60}{k ^{8} - 72 k ^{9}}

Evaluate

(k^{2} - 36) \div (k^{2} - 16 k^{6} \times 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k)

Any expression multiplied by 0 equals 0

(k^{2} - 36) \div (k^{2} - 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k)

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

(k^{2} - 36) \div k^{2} \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k)

Multiply the terms

(k^{2} - 36) \div k^{2} \div (k^{2} - 72 k^{3}) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k)

Multiply

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(k^{2} - 36) \div k^{2} \div (k^{2} - 72 k^{3}) \div (k^{2} - 6 k) (6 k - 60) \div 6 k^{3}

Rewrite the expression

\frac{k ^{2} - 36}{k ^{2}} \div (k^{2} - 72 k^{3}) \div (k^{2} - 6 k) (6 k - 60) \div 6 k^{3}

Divide the terms

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\frac{k ^{2} - 36}{k ^{2} ( k ^{2} - 72 k ^{3} )} \div (k^{2} - 6 k) (6 k - 60) \div 6 k^{3}

Divide the terms

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\frac{k + 6}{k ^{3} ( k ^{2} - 72 k ^{3} )} \times (6 k - 60) \div 6 k^{3}

Multiply the terms

\frac{( k + 6 ) ( 6 k - 60 )}{k ^{3} ( k ^{2} - 72 k ^{3} )} \div 6 k^{3}

Multiply by the reciprocal

\frac{( k + 6 ) ( 6 k - 60 )}{k ^{3} ( k ^{2} - 72 k ^{3} )} \times \frac{1}{6 k ^{3}}

Rewrite the expression

\frac{( k + 6 ) \times 6 ( k - 10 )}{k ^{3} ( k ^{2} - 72 k ^{3} )} \times \frac{1}{6 k ^{3}}

Cancel out the common factor 6

\frac{( k + 6 ) ( k - 10 )}{k ^{3} ( k ^{2} - 72 k ^{3} )} \times \frac{1}{k ^{3}}

Multiply the terms

\frac{( k + 6 ) ( k - 10 )}{k ^{3} ( k ^{2} - 72 k ^{3} ) k ^{3}}

Multiply the terms

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\frac{( k + 6 ) ( k - 10 )}{k ^{6} ( k ^{2} - 72 k ^{3} )}

Multiply the terms

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\frac{k ^{2} - 4 k - 60}{k ^{6} ( k ^{2} - 72 k ^{3} )}

Solution

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\frac{k ^{2} - 4 k - 60}{k ^{8} - 72 k ^{9}}

Show Solution

Find the excluded values

k = 0, k = \frac{1}{72}, k = 6

Evaluate

(k^{2} - 36) \div (k^{2} - 16 k^{6} \times 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k)

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

k^{2} - 16 k^{6} \times 0 = 0 k^{2} - 12 k^{3} \times 6 = 0 k^{2} - 6 k = 0 k^{2} \times 6 k = 0

Solve the equations

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k = 0 k^{2} - 12 k^{3} \times 6 = 0 k^{2} - 6 k = 0 k^{2} \times 6 k = 0

Solve the equations

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k = 0 k = 0 k = \frac{1}{72} k^{2} - 6 k = 0 k^{2} \times 6 k = 0

Solve the equations

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k = 0 k = 0 k = \frac{1}{72} k = 0 k = 6 k^{2} \times 6 k = 0

Solve the equations

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k = 0 k = 0 k = \frac{1}{72} k = 0 k = 6 k = 0

Solution

k = 0, k = \frac{1}{72}, k = 6

Show Solution

Find the roots

k_{1} = - 6, k_{2} = 10

Evaluate

(k^{2} - 36) \div (k^{2} - 16 k^{6} \times 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k)

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

(k^{2} - 36) \div (k^{2} - 16 k^{6} \times 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k) = 0

Find the domain

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(k^{2} - 36) \div (k^{2} - 16 k^{6} \times 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k) = 0, k \in (- \infty, 0) \cup (0, \frac{1}{72}) \cup (\frac{1}{72}, 6) \cup (6, + \infty)

Calculate

(k^{2} - 36) \div (k^{2} - 16 k^{6} \times 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k) = 0

Multiply

More Steps

(k^{2} - 36) \div (k^{2} - 0) \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k) = 0

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

(k^{2} - 36) \div k^{2} \div (k^{2} - 12 k^{3} \times 6) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k) = 0

Multiply the terms

(k^{2} - 36) \div k^{2} \div (k^{2} - 72 k^{3}) \div (k^{2} - 6 k) (6 k - 60) \div (k^{2} \times 6 k) = 0

Multiply

More Steps

(k^{2} - 36) \div k^{2} \div (k^{2} - 72 k^{3}) \div (k^{2} - 6 k) (6 k - 60) \div 6 k^{3} = 0

Rewrite the expression

\frac{k ^{2} - 36}{k ^{2}} \div (k^{2} - 72 k^{3}) \div (k^{2} - 6 k) (6 k - 60) \div 6 k^{3} = 0

Divide the terms

More Steps

\frac{k ^{2} - 36}{k ^{2} ( k ^{2} - 72 k ^{3} )} \div (k^{2} - 6 k) (6 k - 60) \div 6 k^{3} = 0

Divide the terms

More Steps

\frac{k + 6}{k ^{3} ( k ^{2} - 72 k ^{3} )} \times (6 k - 60) \div 6 k^{3} = 0

Multiply the terms

\frac{( k + 6 ) ( 6 k - 60 )}{k ^{3} ( k ^{2} - 72 k ^{3} )} \div 6 k^{3} = 0

Divide the terms

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\frac{( k + 6 ) ( k - 10 )}{k ^{6} ( k ^{2} - 72 k ^{3} )} = 0

Cross multiply

(k + 6) (k - 10) = k^{6} (k^{2} - 72 k^{3}) \times 0

Simplify the equation

(k + 6) (k - 10) = 0

Separate the equation into 2 possible cases

k + 6 = 0 k - 10 = 0

Solve the equation

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k = - 6 k - 10 = 0

Solve the equation

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k = - 6 k = 10

Check if the solution is in the defined range

k = - 6 k = 10, k \in (- \infty, 0) \cup (0, \frac{1}{72}) \cup (\frac{1}{72}, 6) \cup (6, + \infty)

Find the intersection of the solution and the defined range

k = - 6 k = 10

Solution

k_{1} = - 6, k_{2} = 10

Show Solution