Solve for z 3z^2=8

$3 z^{2} = 8$

Step 1

Divide each term in $3 z^{2} = 8$ by $3$ and simplify.

Divide each term in $3 z^{2} = 8$ by $3$ .

$\frac{3 z^{2}}{3} = \frac{8}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 z^{2}}{3} = \frac{8}{3}$

Divide $z^{2}$ by $1$ .

$z^{2} = \frac{8}{3}$

Step 2

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

$z = \pm \sqrt{\frac{8}{3}}$

Step 3

Simplify $\pm \sqrt{\frac{8}{3}}$ .

Rewrite $\sqrt{\frac{8}{3}}$ as $\frac{\sqrt{8}}{\sqrt{3}}$ .

$z = \pm \frac{\sqrt{8}}{\sqrt{3}}$

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

$z = \pm \frac{2 \sqrt{6}}{3}$

Step 4

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.

$z = \frac{2 \sqrt{6}}{3}$

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

$z = - \frac{2 \sqrt{6}}{3}$

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

$z = \frac{2 \sqrt{6}}{3}, - \frac{2 \sqrt{6}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$z = \frac{2 \sqrt{6}}{3}, - \frac{2 \sqrt{6}}{3}$

Decimal Form:

$z = 1.63299316 \dots, - 1.63299316 \dots$

Solve for z 3z^2=8

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