Solve (x^6x)2=7y - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

x = 0

Evaluate

(x^{6} \times x) \times 2 = 7 y

To find the x -intercept,set y =0

(x^{6} \times x) \times 2 = 7 \times 0

Any expression multiplied by 0 equals 0

(x^{6} \times x) \times 2 = 0

Remove the parentheses

x^{6} \times x \times 2 = 0

Multiply

More Steps

Evaluate

x^{6} \times x \times 2

Multiply the terms with the same base by adding their exponents

x^{6 + 1} \times 2

Add the numbers

x^{7} \times 2

Use the commutative property to reorder the terms

2 x^{7}

2 x^{7} = 0

Rewrite the expression

x^{7} = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{7 448 y}{2}

Evaluate

(x^{6} \times x) \times 2 = 7 y

Remove the parentheses

x^{6} \times x \times 2 = 7 y

Multiply

More Steps

Evaluate

x^{6} \times x \times 2

Multiply the terms with the same base by adding their exponents

x^{6 + 1} \times 2

Add the numbers

x^{7} \times 2

Use the commutative property to reorder the terms

2 x^{7}

2 x^{7} = 7 y

Divide both sides

\frac{2 x ^{7}}{2} = \frac{7 y}{2}

Divide the numbers

x^{7} = \frac{7 y}{2}

Take the 7 -th root on both sides of the equation

7 x^{7} = 7 \frac{7 y}{2}

Calculate

x = 7 \frac{7 y}{2}

Solution

More Steps

Evaluate

7 \frac{7 y}{2}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{7 7 y}{7 2}

Multiply by the Conjugate

\frac{7 7 y \times 7 2 ^{6}}{7 2 \times 7 2 ^{6}}

Calculate

\frac{7 7 y \times 7 2 ^{6}}{2}

Calculate

More Steps

Evaluate

7 7 y \times 7 2^{6}

The product of roots with the same index is equal to the root of the product

7 7 y \times 2^{6}

Calculate the product

7 448 y

\frac{7 448 y}{2}

x = \frac{7 448 y}{2}

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

(x^{6} x) 2 = 7 y

Simplify the expression

2 x^{7} = 7 y

To test if the graph of 2 x^{7} = 7 y is symmetry with respect to the origin,substitute -x for x and -y for y

2 (- x)^{7} = 7 (- y)

Evaluate

More Steps

Evaluate

2 (- x)^{7}

Rewrite the expression

2 (- x^{7})

Multiply the numbers

- 2 x^{7}

- 2 x^{7} = 7 (- y)

Evaluate

- 2 x^{7} = - 7 y

Solution

Symmetry with respect to the origin

Show Solution

Rewrite the equation

r = 0 r = \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2} r = - \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2}

Evaluate

(x^{6} \times x) \times 2 = 7 y

Evaluate

More Steps

Evaluate

(x^{6} \times x) \times 2

Remove the parentheses

x^{6} \times x \times 2

Multiply the terms with the same base by adding their exponents

x^{6 + 1} \times 2

Add the numbers

x^{7} \times 2

Use the commutative property to reorder the terms

2 x^{7}

2 x^{7} = 7 y

Move the expression to the left side

2 x^{7} - 7 y = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

2 (cos (θ) \times r)^{7} - 7 sin (θ) \times r = 0

Factor the expression

2 cos^{7} (θ) \times r^{7} - 7 sin (θ) \times r = 0

Factor the expression

r (2 cos^{7} (θ) \times r^{6} - 7 sin (θ)) = 0

When the product of factors equals 0,at least one factor is 0

r = 0 2 cos^{7} (θ) \times r^{6} - 7 sin (θ) = 0

Solution

More Steps

Factor the expression

2 cos^{7} (θ) \times r^{6} - 7 sin (θ) = 0

Subtract the terms

2 cos^{7} (θ) \times r^{6} - 7 sin (θ) - (- 7 sin (θ)) = 0 - (- 7 sin (θ))

Evaluate

2 cos^{7} (θ) \times r^{6} = 7 sin (θ)

Divide the terms

r^{6} = \frac{7 sin ( θ )}{2 cos ^{7} ( θ )}

Simplify the expression

r^{6} = \frac{7 sin ( θ ) sec ^{7} ( θ )}{2}

Evaluate the power

r = \pm 6 \frac{7 sin ( θ ) sec ^{7} ( θ )}{2}

Simplify the expression

More Steps

Evaluate

6 \frac{7 sin ( θ ) sec ^{7} ( θ )}{2}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{6 7 sin ( θ ) sec ^{7} ( θ )}{6 2}

Simplify the radical expression

\frac{6 7 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{6 2}

Multiply by the Conjugate

\frac{6 7 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣ \times 6 2 ^{5}}{6 2 \times 6 2 ^{5}}

Calculate

\frac{6 7 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣ \times 6 2 ^{5}}{2}

Calculate the product

\frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2}

r = \pm \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2}

Separate into possible cases

r = \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2} r = - \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2}

r = 0 r = \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2} r = - \frac{6 224 sin ( θ ) sec ( θ ) \times ∣ sec ( θ ) ∣}{2}

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = 2 x^{6}

Calculate

(x^{6} x) 2 = 7 y

Simplify the expression

2 x^{7} = 7 y

Take the derivative of both sides

\frac{d}{d x} (2 x^{7}) = \frac{d}{d x} (7 y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{7})

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

2 \times \frac{d}{d x} (x^{7})

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

2 \times 7 x^{6}

Multiply the terms

14 x^{6}

14 x^{6} = \frac{d}{d x} (7 y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (7 y)

Use differentiation rules

\frac{d}{d y} (7 y) \times \frac{d y}{d x}

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d y} (7 y)

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

7 \times \frac{d}{d y} (y)

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

7 \times 1

Any expression multiplied by 1 remains the same

7

7 \frac{d y}{d x}

14 x^{6} = 7 \frac{d y}{d x}

Swap the sides of the equation

7 \frac{d y}{d x} = 14 x^{6}

Divide both sides

\frac{7 \frac{d y}{d x}}{7} = \frac{14 x ^{6}}{7}

Divide the numbers

\frac{d y}{d x} = \frac{14 x ^{6}}{7}

Solution

More Steps

Evaluate

\frac{14 x ^{6}}{7}

Reduce the numbers

\frac{2 x ^{6}}{1}

Calculate

2 x^{6}

\frac{d y}{d x} = 2 x^{6}

Show Solution

Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

\frac{d ^{2} y}{d x ^{2}} = 12 x^{5}

Calculate

(x^{6} x) 2 = 7 y

Simplify the expression

2 x^{7} = 7 y

Take the derivative of both sides

\frac{d}{d x} (2 x^{7}) = \frac{d}{d x} (7 y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 x^{7})

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

2 \times \frac{d}{d x} (x^{7})

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

2 \times 7 x^{6}

Multiply the terms

14 x^{6}

14 x^{6} = \frac{d}{d x} (7 y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (7 y)

Use differentiation rules

\frac{d}{d y} (7 y) \times \frac{d y}{d x}

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d y} (7 y)

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

7 \times \frac{d}{d y} (y)

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

7 \times 1

Any expression multiplied by 1 remains the same

7

7 \frac{d y}{d x}

14 x^{6} = 7 \frac{d y}{d x}

Swap the sides of the equation

7 \frac{d y}{d x} = 14 x^{6}

Divide both sides

\frac{7 \frac{d y}{d x}}{7} = \frac{14 x ^{6}}{7}

Divide the numbers

\frac{d y}{d x} = \frac{14 x ^{6}}{7}

Divide the numbers

More Steps

Evaluate

\frac{14 x ^{6}}{7}

Reduce the numbers

\frac{2 x ^{6}}{1}

Calculate

2 x^{6}

\frac{d y}{d x} = 2 x^{6}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (2 x^{6})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (2 x^{6})

Simplify

\frac{d ^{2} y}{d x ^{2}} = 2 \times \frac{d}{d x} (x^{6})

Rewrite the expression

\frac{d ^{2} y}{d x ^{2}} = 2 \times 6 x^{5}

Solution

\frac{d ^{2} y}{d x ^{2}} = 12 x^{5}

Show Solution

Function

Solve the equation

Testing for symmetry

Rewrite the equation

Find the first derivative

Find the second derivative

Graph