Question
Function
Evaluate the derivative
Find the domain
Find the x-intercept/zero
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y′=8192x7
Evaluate
y=4x2×32x6×8
Simplify
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Evaluate
4x2×32x6×8
Multiply the terms
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Evaluate
4×32×8
Multiply the terms
128×8
Multiply the numbers
1024
1024x2×x6
Multiply the terms with the same base by adding their exponents
1024x2+6
Add the numbers
1024x8
y=1024x8
Take the derivative of both sides
y′=dxd(1024x8)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
y′=1024×dxd(x8)
Use dxdxn=nxn−1 to find derivative
y′=1024×8x7
Solution
y′=8192x7
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Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Not symmetry with respect to the origin
Evaluate
y=4x232x68
Simplify the expression
y=1024x8
To test if the graph of y=1024x8 is symmetry with respect to the origin,substitute -x for x and -y for y
−y=1024(−x)8
Simplify
−y=1024x8
Change the signs both sides
y=−1024x8
Solution
Not symmetry with respect to the origin
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Solve the equation
Solve for x
Solve for y
x=4864yx=−4864y
Evaluate
y=4x2×32x6×8
Simplify
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Evaluate
4x2×32x6×8
Multiply the terms
More Steps

Evaluate
4×32×8
Multiply the terms
128×8
Multiply the numbers
1024
1024x2×x6
Multiply the terms with the same base by adding their exponents
1024x2+6
Add the numbers
1024x8
y=1024x8
Swap the sides of the equation
1024x8=y
Divide both sides
10241024x8=1024y
Divide the numbers
x8=1024y
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±81024y
Simplify the expression
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Evaluate
81024y
To take a root of a fraction,take the root of the numerator and denominator separately
810248y
Simplify the radical expression
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Evaluate
81024
Write the expression as a product where the root of one of the factors can be evaluated
8256×4
Write the number in exponential form with the base of 2
828×4
The root of a product is equal to the product of the roots of each factor
828×84
Reduce the index of the radical and exponent with 8
284
Simplify the root
242
2428y
Multiply by the Conjugate
242×4238y×423
Calculate
2×28y×423
Calculate
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Evaluate
8y×423
Use na=mnam to expand the expression
8y×826
The product of roots with the same index is equal to the root of the product
8y×26
Calculate the product
826y
2×2826y
Calculate
4826y
Calculate
4864y
x=±4864y
Solution
x=4864yx=−4864y
Show Solution

Rewrite the equation
r=0r=278cos(θ)×cos(θ)7sin(θ)
Evaluate
y=4x2×32x6×8
Simplify
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Evaluate
4x2×32x6×8
Multiply the terms
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Evaluate
4×32×8
Multiply the terms
128×8
Multiply the numbers
1024
1024x2×x6
Multiply the terms with the same base by adding their exponents
1024x2+6
Add the numbers
1024x8
y=1024x8
Move the expression to the left side
y−1024x8=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
sin(θ)×r−1024(cos(θ)×r)8=0
Factor the expression
−1024cos8(θ)×r8+sin(θ)×r=0
Factor the expression
r(−1024cos8(θ)×r7+sin(θ))=0
When the product of factors equals 0,at least one factor is 0
r=0−1024cos8(θ)×r7+sin(θ)=0
Solution
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Factor the expression
−1024cos8(θ)×r7+sin(θ)=0
Subtract the terms
−1024cos8(θ)×r7+sin(θ)−sin(θ)=0−sin(θ)
Evaluate
−1024cos8(θ)×r7=−sin(θ)
Divide the terms
r7=1024cos8(θ)sin(θ)
Simplify the expression
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Evaluate
71024cos8(θ)sin(θ)
To take a root of a fraction,take the root of the numerator and denominator separately
71024cos8(θ)7sin(θ)
Simplify the radical expression
278cos(θ)×cos(θ)7sin(θ)
r=278cos(θ)×cos(θ)7sin(θ)
r=0r=278cos(θ)×cos(θ)7sin(θ)
Show Solution
