Question
Simplify the expression
k8−72k9k2−4k−60
Evaluate
(k2−36)÷(k2−16k6×0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)
Any expression multiplied by 0 equals 0
(k2−36)÷(k2−0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)
Removing 0 doesn't change the value,so remove it from the expression
(k2−36)÷k2÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)
Multiply the terms
(k2−36)÷k2÷(k2−72k3)÷(k2−6k)(6k−60)÷(k2×6k)
Multiply
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Multiply the terms
k2×6k
Multiply the terms with the same base by adding their exponents
k2+1×6
Add the numbers
k3×6
Use the commutative property to reorder the terms
6k3
(k2−36)÷k2÷(k2−72k3)÷(k2−6k)(6k−60)÷6k3
Rewrite the expression
k2k2−36÷(k2−72k3)÷(k2−6k)(6k−60)÷6k3
Divide the terms
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Evaluate
k2k2−36÷(k2−72k3)
Multiply by the reciprocal
k2k2−36×k2−72k31
Multiply the terms
k2(k2−72k3)k2−36
k2(k2−72k3)k2−36÷(k2−6k)(6k−60)÷6k3
Divide the terms
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Evaluate
k2(k2−72k3)k2−36÷(k2−6k)
Multiply by the reciprocal
k2(k2−72k3)k2−36×k2−6k1
Rewrite the expression
k2(k2−72k3)(k−6)(k+6)×k2−6k1
Rewrite the expression
k2(k2−72k3)(k−6)(k+6)×k(k−6)1
Cancel out the common factor k−6
k2(k2−72k3)k+6×k1
Multiply the terms
k2(k2−72k3)kk+6
Multiply the terms
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Evaluate
k2×k
Use the product rule an×am=an+m to simplify the expression
k2+1
Add the numbers
k3
k3(k2−72k3)k+6
k3(k2−72k3)k+6×(6k−60)÷6k3
Multiply the terms
k3(k2−72k3)(k+6)(6k−60)÷6k3
Multiply by the reciprocal
k3(k2−72k3)(k+6)(6k−60)×6k31
Rewrite the expression
k3(k2−72k3)(k+6)×6(k−10)×6k31
Cancel out the common factor 6
k3(k2−72k3)(k+6)(k−10)×k31
Multiply the terms
k3(k2−72k3)k3(k+6)(k−10)
Multiply the terms
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Evaluate
k3×k3
Use the product rule an×am=an+m to simplify the expression
k3+3
Add the numbers
k6
k6(k2−72k3)(k+6)(k−10)
Multiply the terms
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Evaluate
(k+6)(k−10)
Apply the distributive property
k×k−k×10+6k−6×10
Multiply the terms
k2−k×10+6k−6×10
Use the commutative property to reorder the terms
k2−10k+6k−6×10
Multiply the numbers
k2−10k+6k−60
Add the terms
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Evaluate
−10k+6k
Collect like terms by calculating the sum or difference of their coefficients
(−10+6)k
Add the numbers
−4k
k2−4k−60
k6(k2−72k3)k2−4k−60
Solution
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Evaluate
k6(k2−72k3)
Apply the distributive property
k6×k2−k6×72k3
Multiply the terms
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Evaluate
k6×k2
Use the product rule an×am=an+m to simplify the expression
k6+2
Add the numbers
k8
k8−k6×72k3
Multiply the terms
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Evaluate
k6×72k3
Use the commutative property to reorder the terms
72k6×k3
Multiply the terms
72k9
k8−72k9
k8−72k9k2−4k−60
Show Solution

Find the excluded values
k=0,k=721,k=6
Evaluate
(k2−36)÷(k2−16k6×0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)
To find the excluded values,set the denominators equal to 0
k2−16k6×0=0k2−12k3×6=0k2−6k=0k2×6k=0
Solve the equations
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Evaluate
k2−16k6×0=0
Simplify
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Evaluate
k2−16k6×0
Any expression multiplied by 0 equals 0
k2−0
Removing 0 doesn't change the value,so remove it from the expression
k2
k2=0
The only way a power can be 0 is when the base equals 0
k=0
k=0k2−12k3×6=0k2−6k=0k2×6k=0
Solve the equations
More Steps

Evaluate
k2−12k3×6=0
Multiply the terms
k2−72k3=0
Factor the expression
k2(1−72k)=0
Separate the equation into 2 possible cases
k2=01−72k=0
The only way a power can be 0 is when the base equals 0
k=01−72k=0
Solve the equation
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Evaluate
1−72k=0
Move the constant to the right-hand side and change its sign
−72k=0−1
Removing 0 doesn't change the value,so remove it from the expression
−72k=−1
Change the signs on both sides of the equation
72k=1
Divide both sides
7272k=721
Divide the numbers
k=721
k=0k=721
k=0k=0k=721k2−6k=0k2×6k=0
Solve the equations
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Evaluate
k2−6k=0
Factor the expression
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Evaluate
k2−6k
Rewrite the expression
k×k−k×6
Factor out k from the expression
k(k−6)
k(k−6)=0
When the product of factors equals 0,at least one factor is 0
k=0k−6=0
Solve the equation for k
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Evaluate
k−6=0
Move the constant to the right-hand side and change its sign
k=0+6
Removing 0 doesn't change the value,so remove it from the expression
k=6
k=0k=6
k=0k=0k=721k=0k=6k2×6k=0
Solve the equations
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Evaluate
k2×6k=0
Multiply
More Steps

Evaluate
k2×6k
Multiply the terms with the same base by adding their exponents
k2+1×6
Add the numbers
k3×6
Use the commutative property to reorder the terms
6k3
6k3=0
Rewrite the expression
k3=0
The only way a power can be 0 is when the base equals 0
k=0
k=0k=0k=721k=0k=6k=0
Solution
k=0,k=721,k=6
Show Solution

Find the roots
k1=−6,k2=10
Evaluate
(k2−36)÷(k2−16k6×0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)
To find the roots of the expression,set the expression equal to 0
(k2−36)÷(k2−16k6×0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)=0
Find the domain
More Steps

Evaluate
⎩⎨⎧k2−16k6×0=0k2−12k3×6=0k2−6k=0k2×6k=0
Calculate
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Evaluate
k2−16k6×0=0
Simplify
k2=0
The only way a power can not be 0 is when the base not equals 0
k=0
⎩⎨⎧k=0k2−12k3×6=0k2−6k=0k2×6k=0
Calculate
More Steps

Evaluate
k2−12k3×6=0
Multiply the terms
k2−72k3=0
Factor the expression
k2(1−72k)=0
Apply the zero product property
{k2=01−72k=0
The only way a power can not be 0 is when the base not equals 0
{k=01−72k=0
Solve the inequality
{k=0k=721
Find the intersection
k∈(−∞,0)∪(0,721)∪(721,+∞)
⎩⎨⎧k=0k∈(−∞,0)∪(0,721)∪(721,+∞)k2−6k=0k2×6k=0
Calculate
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Evaluate
k2−6k=0
Add the same value to both sides
k2−6k+9=9
Evaluate
(k−3)2=9
Take the root of both sides of the equation and remember to use both positive and negative roots
k−3=±9
Simplify the expression
k−3=±3
Separate the inequality into 2 possible cases
{k−3=3k−3=−3
Calculate
{k=6k−3=−3
Cancel equal terms on both sides of the expression
{k=6k=0
Find the intersection
k∈(−∞,0)∪(0,6)∪(6,+∞)
⎩⎨⎧k=0k∈(−∞,0)∪(0,721)∪(721,+∞)k∈(−∞,0)∪(0,6)∪(6,+∞)k2×6k=0
Calculate
More Steps

Evaluate
k2×6k=0
Multiply
6k3=0
Rewrite the expression
k3=0
The only way a power can not be 0 is when the base not equals 0
k=0
⎩⎨⎧k=0k∈(−∞,0)∪(0,721)∪(721,+∞)k∈(−∞,0)∪(0,6)∪(6,+∞)k=0
Simplify
⎩⎨⎧k=0k∈(−∞,0)∪(0,721)∪(721,+∞)k∈(−∞,0)∪(0,6)∪(6,+∞)
Find the intersection
k∈(−∞,0)∪(0,721)∪(721,6)∪(6,+∞)
(k2−36)÷(k2−16k6×0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)=0,k∈(−∞,0)∪(0,721)∪(721,6)∪(6,+∞)
Calculate
(k2−36)÷(k2−16k6×0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)=0
Multiply
More Steps

Multiply the terms
16k6×0
Any expression multiplied by 0 equals 0
0×k6
Any expression multiplied by 0 equals 0
0
(k2−36)÷(k2−0)÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)=0
Removing 0 doesn't change the value,so remove it from the expression
(k2−36)÷k2÷(k2−12k3×6)÷(k2−6k)(6k−60)÷(k2×6k)=0
Multiply the terms
(k2−36)÷k2÷(k2−72k3)÷(k2−6k)(6k−60)÷(k2×6k)=0
Multiply
More Steps

Multiply the terms
k2×6k
Multiply the terms with the same base by adding their exponents
k2+1×6
Add the numbers
k3×6
Use the commutative property to reorder the terms
6k3
(k2−36)÷k2÷(k2−72k3)÷(k2−6k)(6k−60)÷6k3=0
Rewrite the expression
k2k2−36÷(k2−72k3)÷(k2−6k)(6k−60)÷6k3=0
Divide the terms
More Steps

Evaluate
k2k2−36÷(k2−72k3)
Multiply by the reciprocal
k2k2−36×k2−72k31
Multiply the terms
k2(k2−72k3)k2−36
k2(k2−72k3)k2−36÷(k2−6k)(6k−60)÷6k3=0
Divide the terms
More Steps

Evaluate
k2(k2−72k3)k2−36÷(k2−6k)
Multiply by the reciprocal
k2(k2−72k3)k2−36×k2−6k1
Rewrite the expression
k2(k2−72k3)(k−6)(k+6)×k2−6k1
Rewrite the expression
k2(k2−72k3)(k−6)(k+6)×k(k−6)1
Cancel out the common factor k−6
k2(k2−72k3)k+6×k1
Multiply the terms
k2(k2−72k3)kk+6
Multiply the terms
More Steps

Evaluate
k2×k
Use the product rule an×am=an+m to simplify the expression
k2+1
Add the numbers
k3
k3(k2−72k3)k+6
k3(k2−72k3)k+6×(6k−60)÷6k3=0
Multiply the terms
k3(k2−72k3)(k+6)(6k−60)÷6k3=0
Divide the terms
More Steps

Evaluate
k3(k2−72k3)(k+6)(6k−60)÷6k3
Multiply by the reciprocal
k3(k2−72k3)(k+6)(6k−60)×6k31
Rewrite the expression
k3(k2−72k3)(k+6)×6(k−10)×6k31
Cancel out the common factor 6
k3(k2−72k3)(k+6)(k−10)×k31
Multiply the terms
k3(k2−72k3)k3(k+6)(k−10)
Multiply the terms
More Steps

Evaluate
k3×k3
Use the product rule an×am=an+m to simplify the expression
k3+3
Add the numbers
k6
k6(k2−72k3)(k+6)(k−10)
k6(k2−72k3)(k+6)(k−10)=0
Cross multiply
(k+6)(k−10)=k6(k2−72k3)×0
Simplify the equation
(k+6)(k−10)=0
Separate the equation into 2 possible cases
k+6=0k−10=0
Solve the equation
More Steps

Evaluate
k+6=0
Move the constant to the right-hand side and change its sign
k=0−6
Removing 0 doesn't change the value,so remove it from the expression
k=−6
k=−6k−10=0
Solve the equation
More Steps

Evaluate
k−10=0
Move the constant to the right-hand side and change its sign
k=0+10
Removing 0 doesn't change the value,so remove it from the expression
k=10
k=−6k=10
Check if the solution is in the defined range
k=−6k=10,k∈(−∞,0)∪(0,721)∪(721,6)∪(6,+∞)
Find the intersection of the solution and the defined range
k=−6k=10
Solution
k1=−6,k2=10
Show Solution
