Find the Roots (Zeros) 9x^4-16=0

$9 x^{4} - 16 = 0$

Step 1

Add $16$ to both sides of the equation.

$9 x^{4} = 16$

Step 2

Divide each term in $9 x^{4} = 16$ by $9$ and simplify.

Divide each term in $9 x^{4} = 16$ by $9$ .

$\frac{9 x^{4}}{9} = \frac{16}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x^{4}}{9} = \frac{16}{9}$

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

$x^{4} = \frac{16}{9}$

Step 3

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

$x = \pm \sqrt[4]{\frac{16}{9}}$

Step 4

Simplify $\pm \sqrt[4]{\frac{16}{9}}$ .

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt[4]{\frac{4^{2}}{9}}$

Rewrite $9$ as $3^{2}$ .

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

Rewrite $\frac{4^{2}}{3^{2}}$ as ${(\frac{4}{3})}^{2}$ .

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

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

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

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

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

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

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

Simplify the numerator.

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

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

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

$x = \pm \frac{2}{\sqrt{3}}$

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{3}}{3}$

Step 5

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.

$x = \frac{2 \sqrt{3}}{3}$

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

$x = - \frac{2 \sqrt{3}}{3}$

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

$x = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \sqrt{3}}{3}, - \frac{2 \sqrt{3}}{3}$

Decimal Form:

$x = 1.15470053 \dots, - 1.15470053 \dots$

Step 7

Find the Roots (Zeros) 9x^4-16=0

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