Question
Solve the equation
Solve for x
Solve for y
x=0
Evaluate
x2y2×16x4y4×3=0
Multiply
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Evaluate
x2y2×16x4y4×3
Multiply the terms with the same base by adding their exponents
x2+4y2×16y4×3
Add the numbers
x6y2×16y4×3
Multiply the terms with the same base by adding their exponents
x6y2+4×16×3
Add the numbers
x6y6×16×3
Multiply the terms
x6y6×48
Use the commutative property to reorder the terms
48x6y6
48x6y6=0
Rewrite the expression
48y6x6=0
Rewrite the expression
x6=0
Solution
x=0
Show Solution

Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Symmetry with respect to the origin
Evaluate
x2y2×16x4y4×3=0
Multiply
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Evaluate
x2y2×16x4y4×3
Multiply the terms with the same base by adding their exponents
x2+4y2×16y4×3
Add the numbers
x6y2×16y4×3
Multiply the terms with the same base by adding their exponents
x6y2+4×16×3
Add the numbers
x6y6×16×3
Multiply the terms
x6y6×48
Use the commutative property to reorder the terms
48x6y6
48x6y6=0
To test if the graph of 48x6y6=0 is symmetry with respect to the origin,substitute -x for x and -y for y
48(−x)6(−y)6=0
Evaluate
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Evaluate
48(−x)6(−y)6
Multiply the terms
48x6(−y)6
Multiply the terms
48x6y6
48x6y6=0
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=0θ=2kπ,k∈Z
Evaluate
x2y2×16x4y4×3=0
Evaluate
More Steps

Evaluate
x2y2×16x4y4×3
Multiply the terms with the same base by adding their exponents
x2+4y2×16y4×3
Add the numbers
x6y2×16y4×3
Multiply the terms with the same base by adding their exponents
x6y2+4×16×3
Add the numbers
x6y6×16×3
Multiply the terms
x6y6×48
Use the commutative property to reorder the terms
48x6y6
48x6y6=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
48(cos(θ)×r)6(sin(θ)×r)6=0
Factor the expression
48cos6(θ)sin6(θ)×r12=0
Separate into possible cases
r12=048cos6(θ)sin6(θ)=0
Evaluate
r=048cos6(θ)sin6(θ)=0
Solution
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Evaluate
48cos6(θ)sin6(θ)=0
Elimination the left coefficient
cos6(θ)sin6(θ)=0
Separate the equation into 2 possible cases
cos6(θ)=0sin6(θ)=0
Solve the equation
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Evaluate
cos6(θ)=0
The only way a power can be 0 is when the base equals 0
cos(θ)=0
Use the inverse trigonometric function
θ=arccos(0)
Calculate
θ=2π
Add the period of kπ,k∈Z to find all solutions
θ=2π+kπ,k∈Z
θ=2π+kπ,k∈Zsin6(θ)=0
Solve the equation
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Evaluate
sin6(θ)=0
The only way a power can be 0 is when the base equals 0
sin(θ)=0
Use the inverse trigonometric function
θ=arcsin(0)
Calculate
θ=0
Add the period of kπ,k∈Z to find all solutions
θ=kπ,k∈Z
θ=2π+kπ,k∈Zθ=kπ,k∈Z
Find the union
θ=2kπ,k∈Z
r=0θ=2kπ,k∈Z
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−xy
Calculate
x2y216x4y43=0
Simplify the expression
48x6y6=0
Take the derivative of both sides
dxd(48x6y6)=dxd(0)
Calculate the derivative
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Evaluate
dxd(48x6y6)
Use differentiation rules
dxd(48x6)×y6+48x6×dxd(y6)
Evaluate the derivative
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Evaluate
dxd(48x6)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
48×dxd(x6)
Use dxdxn=nxn−1 to find derivative
48×6x5
Multiply the terms
288x5
288x5y6+48x6×dxd(y6)
Evaluate the derivative
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Evaluate
dxd(y6)
Use differentiation rules
dyd(y6)×dxdy
Use dxdxn=nxn−1 to find derivative
6y5dxdy
288x5y6+288x6y5dxdy
288x5y6+288x6y5dxdy=dxd(0)
Calculate the derivative
288x5y6+288x6y5dxdy=0
Move the expression to the right-hand side and change its sign
288x6y5dxdy=0−288x5y6
Removing 0 doesn't change the value,so remove it from the expression
288x6y5dxdy=−288x5y6
Divide both sides
288x6y5288x6y5dxdy=288x6y5−288x5y6
Divide the numbers
dxdy=288x6y5−288x5y6
Solution
More Steps

Evaluate
288x6y5−288x5y6
Cancel out the common factor 288
x6y5−x5y6
Reduce the fraction
More Steps

Evaluate
x6x5
Use the product rule aman=an−m to simplify the expression
x6−51
Subtract the terms
x11
Simplify
x1
xy5−y6
Reduce the fraction
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Evaluate
y5y6
Use the product rule aman=an−m to simplify the expression
y6−5
Subtract the terms
y1
Simplify
y
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=x22y
Calculate
x2y216x4y43=0
Simplify the expression
48x6y6=0
Take the derivative of both sides
dxd(48x6y6)=dxd(0)
Calculate the derivative
More Steps

Evaluate
dxd(48x6y6)
Use differentiation rules
dxd(48x6)×y6+48x6×dxd(y6)
Evaluate the derivative
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Evaluate
dxd(48x6)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
48×dxd(x6)
Use dxdxn=nxn−1 to find derivative
48×6x5
Multiply the terms
288x5
288x5y6+48x6×dxd(y6)
Evaluate the derivative
More Steps

Evaluate
dxd(y6)
Use differentiation rules
dyd(y6)×dxdy
Use dxdxn=nxn−1 to find derivative
6y5dxdy
288x5y6+288x6y5dxdy
288x5y6+288x6y5dxdy=dxd(0)
Calculate the derivative
288x5y6+288x6y5dxdy=0
Move the expression to the right-hand side and change its sign
288x6y5dxdy=0−288x5y6
Removing 0 doesn't change the value,so remove it from the expression
288x6y5dxdy=−288x5y6
Divide both sides
288x6y5288x6y5dxdy=288x6y5−288x5y6
Divide the numbers
dxdy=288x6y5−288x5y6
Divide the numbers
More Steps

Evaluate
288x6y5−288x5y6
Cancel out the common factor 288
x6y5−x5y6
Reduce the fraction
More Steps

Evaluate
x6x5
Use the product rule aman=an−m to simplify the expression
x6−51
Subtract the terms
x11
Simplify
x1
xy5−y6
Reduce the fraction
More Steps

Evaluate
y5y6
Use the product rule aman=an−m to simplify the expression
y6−5
Subtract the terms
y1
Simplify
y
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Take the derivative of both sides
dxd(dxdy)=dxd(−xy)
Calculate the derivative
dx2d2y=dxd(−xy)
Use differentiation rules
dx2d2y=−x2dxd(y)×x−y×dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dx2d2y=−x2dxdy×x−y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x2dxdy×x−y×1
Use the commutative property to reorder the terms
dx2d2y=−x2xdxdy−y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x2xdxdy−y
Use equation dxdy=−xy to substitute
dx2d2y=−x2x(−xy)−y
Solution
More Steps

Calculate
−x2x(−xy)−y
Multiply the terms
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Evaluate
x(−xy)
Multiplying or dividing an odd number of negative terms equals a negative
−x×xy
Cancel out the common factor x
−1×y
Multiply the terms
−y
−x2−y−y
Subtract the terms
More Steps

Simplify
−y−y
Collect like terms by calculating the sum or difference of their coefficients
(−1−1)y
Subtract the numbers
−2y
−x2−2y
Divide the terms
−(−x22y)
Calculate
x22y
dx2d2y=x22y
Show Solution
