Solve y = x^48x - ScanMath

Question

Function

Find the inverse

Evaluate the derivative

Find the domain

f^{- 1} (x) = \frac{5 4 x}{2}

Evaluate

y = x^{4} \times 8 x

Simplify

More Steps

Evaluate

x^{4} \times 8 x

Multiply the terms with the same base by adding their exponents

x^{4 + 1} \times 8

Add the numbers

x^{5} \times 8

Use the commutative property to reorder the terms

8 x^{5}

y = 8 x^{5}

Interchange x and y

x = 8 y^{5}

Swap the sides of the equation

8 y^{5} = x

Divide both sides

\frac{8 y ^{5}}{8} = \frac{x}{8}

Divide the numbers

y^{5} = \frac{x}{8}

Take the 5 -th root on both sides of the equation

5 y^{5} = 5 \frac{x}{8}

Calculate

y = 5 \frac{x}{8}

Simplify the root

More Steps

Evaluate

5 \frac{x}{8}

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

\frac{5 x}{5 8}

Multiply by the Conjugate

\frac{5 x \times 5 8 ^{4}}{5 8 \times 5 8 ^{4}}

Calculate

\frac{5 x \times 5 8 ^{4}}{2 ^{3}}

Calculate

More Steps

Evaluate

5 x \times 5 8^{4}

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

5 x \times 8^{4}

Calculate the product

5 8^{4} x

Rewrite the exponent as a sum

5 2^{10 + 2} x

Use a^{m + n} = a^{m} \times a^{n} to expand the expression

5 2^{10} \times 2^{2} x

The root of a product is equal to the product of the roots of each factor

5 2^{10} \times 5 2^{2} x

Reduce the index of the radical and exponent with 5

4 5 4 x

\frac{4 5 4 x}{2 ^{3}}

Divide the terms

More Steps

Evaluate

\frac{4}{2 ^{3}}

Rewrite the expression

\frac{2 ^{2}}{2 ^{3}}

Use the product rule \frac{a ^{n}}{a ^{m}} = a^{n - m} to simplify the expression

\frac{1}{2 ^{3 - 2}}

Subtract the terms

\frac{1}{2 ^{1}}

Simplify

\frac{1}{2}

\frac{5 4 x}{2}

y = \frac{5 4 x}{2}

Solution

f^{- 1} (x) = \frac{5 4 x}{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

y = x^{4} 8 x

Simplify the expression

y = 8 x^{5}

To test if the graph of y = 8 x^{5} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 8 (- x)^{5}

Simplify

More Steps

Evaluate

8 (- x)^{5}

Rewrite the expression

8 (- x^{5})

Multiply the numbers

- 8 x^{5}

- y = - 8 x^{5}

Change the signs both sides

y = 8 x^{5}

Solution

Symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{5 4 y}{2}

Evaluate

y = x^{4} \times 8 x

Simplify

More Steps

Evaluate

x^{4} \times 8 x

Multiply the terms with the same base by adding their exponents

x^{4 + 1} \times 8

Add the numbers

x^{5} \times 8

Use the commutative property to reorder the terms

8 x^{5}

y = 8 x^{5}

Swap the sides of the equation

8 x^{5} = y

Divide both sides

\frac{8 x ^{5}}{8} = \frac{y}{8}

Divide the numbers

x^{5} = \frac{y}{8}

Take the 5 -th root on both sides of the equation

5 x^{5} = 5 \frac{y}{8}

Calculate

x = 5 \frac{y}{8}

Solution

More Steps

Evaluate

5 \frac{y}{8}

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

\frac{5 y}{5 8}

Multiply by the Conjugate

\frac{5 y \times 5 8 ^{4}}{5 8 \times 5 8 ^{4}}

Calculate

\frac{5 y \times 5 8 ^{4}}{2 ^{3}}

Calculate

More Steps

Evaluate

5 y \times 5 8^{4}

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

5 y \times 8^{4}

Calculate the product

5 8^{4} y

Rewrite the exponent as a sum

5 2^{10 + 2} y

Use a^{m + n} = a^{m} \times a^{n} to expand the expression

5 2^{10} \times 2^{2} y

The root of a product is equal to the product of the roots of each factor

5 2^{10} \times 5 2^{2} y

Reduce the index of the radical and exponent with 5

4 5 4 y

\frac{4 5 4 y}{2 ^{3}}

Divide the terms

More Steps

Evaluate

\frac{4}{2 ^{3}}

Rewrite the expression

\frac{2 ^{2}}{2 ^{3}}

Use the product rule \frac{a ^{n}}{a ^{m}} = a^{n - m} to simplify the expression

\frac{1}{2 ^{3 - 2}}

Subtract the terms

\frac{1}{2 ^{1}}

Simplify

\frac{1}{2}

\frac{5 4 y}{2}

x = \frac{5 4 y}{2}

Show Solution

Rewrite the equation

r = 0 r = 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )} r = - 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )}

Evaluate

y = x^{4} \times 8 x

Simplify

More Steps

Evaluate

x^{4} \times 8 x

Multiply the terms with the same base by adding their exponents

x^{4 + 1} \times 8

Add the numbers

x^{5} \times 8

Use the commutative property to reorder the terms

8 x^{5}

y = 8 x^{5}

Move the expression to the left side

y - 8 x^{5} = 0

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

sin (θ) \times r - 8 (cos (θ) \times r)^{5} = 0

Factor the expression

- 8 cos^{5} (θ) \times r^{5} + sin (θ) \times r = 0

Factor the expression

r (- 8 cos^{5} (θ) \times r^{4} + sin (θ)) = 0

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

r = 0 - 8 cos^{5} (θ) \times r^{4} + sin (θ) = 0

Solution

More Steps

Factor the expression

- 8 cos^{5} (θ) \times r^{4} + sin (θ) = 0

Subtract the terms

- 8 cos^{5} (θ) \times r^{4} + sin (θ) - sin (θ) = 0 - sin (θ)

Evaluate

- 8 cos^{5} (θ) \times r^{4} = - sin (θ)

Divide the terms

r^{4} = \frac{sin ( θ )}{8 cos ^{5} ( θ )}

Evaluate the power

r = \pm 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )}

Separate into possible cases

r = 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )} r = - 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )}

r = 0 r = 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )} r = - 4 \frac{sin ( θ )}{8 cos ^{5} ( θ )}

Show Solution

Function

Testing for symmetry

Solve the equation

Rewrite the equation

Graph