Solve x^2y^4z=21 - ScanMath

Question

Solve the equation

Solve for x

Solve for y

Solve for z

x = \frac{21 z}{y ^{2} ∣ z ∣} x = - \frac{21 z}{y ^{2} ∣ z ∣}

Evaluate

x^{2} y^{4} z = 21

Rewrite the expression

y^{4} z x^{2} = 21

Divide both sides

\frac{y ^{4} z x ^{2}}{y ^{4} z} = \frac{21}{y ^{4} z}

Divide the numbers

x^{2} = \frac{21}{y ^{4} z}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm \frac{21}{y ^{4} z}

Simplify the expression

More Steps

Evaluate

\frac{21}{y ^{4} z}

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

\frac{21}{y ^{4} z}

Simplify the radical expression

More Steps

Evaluate

y^{4} z

Rewrite the expression

y^{4} \times z

Simplify the root

y^{2} z

\frac{21}{y ^{2} z}

Multiply by the Conjugate

\frac{21 \times z}{y ^{2} z \times z}

Calculate

\frac{21 \times z}{y ^{2} ∣ z ∣}

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

\frac{21 z}{y ^{2} ∣ z ∣}

Calculate

\frac{21 z}{∣ z ∣ \times y ^{2}}

x = \pm \frac{21 z}{∣ z ∣ \times y ^{2}}

Separate the equation into 2 possible cases

x = \frac{21 z}{∣ z ∣ \times y ^{2}} x = - \frac{21 z}{∣ z ∣ \times y ^{2}}

Multiply the terms

x = \frac{21 z}{y ^{2} ∣ z ∣} x = - \frac{21 z}{∣ z ∣ \times y ^{2}}

Solution

x = \frac{21 z}{y ^{2} ∣ z ∣} x = - \frac{21 z}{y ^{2} ∣ z ∣}

Show Solution

Find \frac{\partial z}{\partial x} by differentiating the equation directly

Find \frac{\partial z}{\partial y} by differentiating the equation directly

\frac{\partial z}{\partial x} = - \frac{2 z}{x}

Evaluate

x^{2} y^{4} z = 21

Find \frac{\partial z}{\partial x} by taking the derivative of both sides with respect to x

\frac{\partial}{\partial x} (x^{2} y^{4} z) = \frac{\partial}{\partial x} (21)

Use differentiation rule \frac{\partial}{\partial x} (f (x) \times g (x)) = \frac{\partial}{\partial x} (f (x)) \times g (x) + f (x) \times \frac{\partial}{\partial x} (g (x))

\frac{\partial}{\partial x} (x^{2} y^{4}) z + x^{2} y^{4} \times \frac{\partial}{\partial x} (z) = \frac{\partial}{\partial x} (21)

Evaluate

More Steps

Evaluate

\frac{\partial}{\partial x} (x^{2} y^{4})

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

y^{4} \times \frac{\partial}{\partial x} (x^{2})

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

y^{4} \times 2 x

Multiply the terms

2 x y^{4}

2 x y^{4} z + x^{2} y^{4} \times \frac{\partial}{\partial x} (z) = \frac{\partial}{\partial x} (21)

Evaluate

2 x z y^{4} + x^{2} y^{4} \times \frac{\partial}{\partial x} (z) = \frac{\partial}{\partial x} (21)

Evaluate

2 x z y^{4} + x^{2} y^{4} \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (21)

Find the partial derivative

2 x z y^{4} + x^{2} y^{4} \frac{\partial z}{\partial x} = 0

Move the expression to the right-hand side and change its sign

x^{2} y^{4} \frac{\partial z}{\partial x} = 0 - 2 x z y^{4}

Removing 0 doesn't change the value,so remove it from the expression

x^{2} y^{4} \frac{\partial z}{\partial x} = - 2 x z y^{4}

Divide both sides

\frac{x ^{2} y ^{4} \frac{\partial z}{\partial x}}{x ^{2} y ^{4}} = \frac{- 2 x z y ^{4}}{x ^{2} y ^{4}}

Divide the numbers

\frac{\partial z}{\partial x} = \frac{- 2 x z y ^{4}}{x ^{2} y ^{4}}

Solution

More Steps

Evaluate

\frac{- 2 x z y ^{4}}{x ^{2} y ^{4}}

Reduce the fraction

\frac{- 2 z y ^{4}}{x y ^{4}}

Reduce the fraction

\frac{- 2 z}{x}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{2 z}{x}

\frac{\partial z}{\partial x} = - \frac{2 z}{x}

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