Find the Roots (Zeros) f(x)=4x^5-3x

$f (x) = 4 x^{5} - 3 x$

Step 1

Set $4 x^{5} - 3 x$ equal to $0$ .

$4 x^{5} - 3 x = 0$

Step 2

Solve for $x$ .

Factor $x$ out of $4 x^{5} - 3 x$ .

Factor $x$ out of $4 x^{5}$ .

$x (4 x^{4}) - 3 x = 0$

Factor $x$ out of $- 3 x$ .

$x (4 x^{4}) + x \cdot - 3 = 0$

Factor $x$ out of $x (4 x^{4}) + x \cdot - 3$ .

$x (4 x^{4} - 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 x^{4} - 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $4 x^{4} - 3$ equal to $0$ and solve for $x$ .

Set $4 x^{4} - 3$ equal to $0$ .

$4 x^{4} - 3 = 0$

Solve $4 x^{4} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$4 x^{4} = 3$

Divide each term in $4 x^{4} = 3$ by $4$ and simplify.

Divide each term in $4 x^{4} = 3$ by $4$ .

$\frac{4 x^{4}}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{4}}{4} = \frac{3}{4}$

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

$x^{4} = \frac{3}{4}$

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

$x = \pm \sqrt[4]{\frac{3}{4}}$

Simplify $\pm \sqrt[4]{\frac{3}{4}}$ .

Rewrite $\sqrt[4]{\frac{3}{4}}$ as $\frac{\sqrt[4]{3}}{\sqrt[4]{4}}$ .

$x = \pm \frac{\sqrt[4]{3}}{\sqrt[4]{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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

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

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{\sqrt[4]{3}}{\sqrt{2}}$

Multiply $\frac{\sqrt[4]{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt[4]{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt[4]{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$x = \pm \frac{\sqrt[4]{3} \sqrt{2}}{2}$

Simplify the numerator.

Rewrite the expression using the least common index of $4$ .

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

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

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

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

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

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

Combine using the product rule for radicals.

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

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

$x = \pm \frac{\sqrt[4]{3 \cdot 4}}{2}$

Multiply $3$ by $4$ .

$x = \pm \frac{\sqrt[4]{12}}{2}$

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.

$x = \frac{\sqrt[4]{12}}{2}$

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

$x = - \frac{\sqrt[4]{12}}{2}$

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

$x = \frac{\sqrt[4]{12}}{2}, - \frac{\sqrt[4]{12}}{2}$

The final solution is all the values that make $x (4 x^{4} - 3) = 0$ true.

$x = 0, \frac{\sqrt[4]{12}}{2}, - \frac{\sqrt[4]{12}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{\sqrt[4]{12}}{2}, - \frac{\sqrt[4]{12}}{2}$

Decimal Form:

$x = 0, 0.93060485 \dots, - 0.93060485 \dots$

Step 4

Find the Roots (Zeros) f(x)=4x^5-3x

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