Solve for x y=2x^4-5

$y = 2 x^{4} - 5$

Step 1

Rewrite the equation as $2 x^{4} - 5 = y$ .

$2 x^{4} - 5 = y$

Step 2

Add $5$ to both sides of the equation.

$2 x^{4} = y + 5$

Step 3

Divide each term in $2 x^{4} = y + 5$ by $2$ and simplify.

Divide each term in $2 x^{4} = y + 5$ by $2$ .

$\frac{2 x^{4}}{2} = \frac{y}{2} + \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{4}}{2} = \frac{y}{2} + \frac{5}{2}$

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{y}{2} + \frac{5}{2}$

Step 4

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{\frac{y}{2} + \frac{5}{2}}$

Step 5

Simplify $\pm \sqrt[4]{\frac{y}{2} + \frac{5}{2}}$ .

Combine the numerators over the common denominator.

$x = \pm \sqrt[4]{\frac{y + 5}{2}}$

Rewrite $\sqrt[4]{\frac{y + 5}{2}}$ as $\frac{\sqrt[4]{y + 5}}{\sqrt[4]{2}}$ .

$x = \pm \frac{\sqrt[4]{y + 5}}{\sqrt[4]{2}}$

Multiply $\frac{\sqrt[4]{y + 5}}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$x = \pm \frac{\sqrt[4]{y + 5}}{\sqrt[4]{2}} \cdot \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{y + 5}}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{\sqrt[4]{2} {\sqrt[4]{2}}^{3}}$

Raise $\sqrt[4]{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1} {\sqrt[4]{2}}^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1 + 3}}$

Add $1$ and $3$ .

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{4}}$

Rewrite ${\sqrt[4]{2}}^{4}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2}$ as $2^{\frac{1}{4}}$ .

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{{(2^{\frac{1}{4}})}^{4}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{2^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{2^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{2^{\frac{4}{4}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt[4]{y + 5} {\sqrt[4]{2}}^{3}}{2}$

Simplify the numerator.

Rewrite ${\sqrt[4]{2}}^{3}$ as ${(2^{3})}^{\frac{1}{4}}$ .

$x = \pm \frac{\sqrt[4]{y + 5} \sqrt[4]{2^{3}}}{2}$

Raise $2$ to the power of $3$ .

$x = \pm \frac{\sqrt[4]{y + 5} \sqrt[4]{8}}{2}$

Simplify with factoring out.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt[4]{(y + 5) \cdot 8}}{2}$

Reorder factors in $\pm \frac{\sqrt[4]{(y + 5) \cdot 8}}{2}$ .

$x = \pm \frac{\sqrt[4]{8 (y + 5)}}{2}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt[4]{8 (y + 5)}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt[4]{8 (y + 5)}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt[4]{8 (y + 5)}}{2}$

$x = - \frac{\sqrt[4]{8 (y + 5)}}{2}$

$x = \frac{\sqrt[4]{8 (y + 5)}}{2}$

$x = - \frac{\sqrt[4]{8 (y + 5)}}{2}$

Solve for x y=2x^4-5

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