Precalculus Examples

$\frac{6^{\frac{3}{2}}}{{(\sqrt{2})}^{3}} = t^{\frac{2}{3}}$

Step 1

Rewrite the equation as $t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{{\sqrt{2}}^{3}}$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{{\sqrt{2}}^{3}}$

Step 2

Move the terms containing $t$ to the left side and simplify.

Tap for more steps...

Step 2.1

Simplify the denominator.

Tap for more steps...

Step 2.1.1

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{\sqrt{2^{3}}}$

Step 2.1.2

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{\sqrt{8}}$

Step 2.1.3

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

Tap for more steps...

Step 2.1.3.1

Factor $4$ out of $8$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{\sqrt{4 (2)}}$

Step 2.1.3.2

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{\sqrt{2^{2} \cdot 2}}$

Step 2.1.4

Pull terms out from under the radical.

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{2 \sqrt{2}}$

Step 2.2

Multiply $\frac{6^{\frac{3}{2}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.3.1

Multiply $\frac{6^{\frac{3}{2}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 2.3.2

Move $\sqrt{2}$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 2.3.3

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 2.3.4

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 2.3.5

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 2.3.6

Add $1$ and $1$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 2.3.7

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

Tap for more steps...

Step 2.3.7.1

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 2.3.7.2

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 2.3.7.3

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

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 2.3.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.7.4.1

Cancel the common factor.

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 2.3.7.4.2

Rewrite the expression.

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 \cdot 2}$

Step 2.3.7.5

Evaluate the exponent.

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{2 \cdot 2}$

Step 2.4

Multiply $2$ by $2$ .

$t^{\frac{2}{3}} = \frac{6^{\frac{3}{2}} \sqrt{2}}{4}$

Step 3

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(t^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4

Simplify the exponent.

Tap for more steps...

Step 4.1

Simplify the left side.

Tap for more steps...

Step 4.1.1

Simplify ${(t^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 4.1.1.1

Multiply the exponents in ${(t^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 4.1.1.1.1

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

$t^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.1.1.2.1

Cancel the common factor.

$t^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4.1.1.1.2.2

Rewrite the expression.

$t^{\frac{1}{3} \cdot 3} = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4.1.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.1.1.1.3.1

Cancel the common factor.

$t^{\frac{1}{3} \cdot 3} = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4.1.1.1.3.2

Rewrite the expression.

$t^{1} = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4.1.1.2

Simplify.

$t = \pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $\pm {(\frac{6^{\frac{3}{2}} \sqrt{2}}{4})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.2.1.1.1

Apply the product rule to $\frac{6^{\frac{3}{2}} \sqrt{2}}{4}$ .

$t = \pm \frac{{(6^{\frac{3}{2}} \sqrt{2})}^{\frac{3}{2}}}{4^{\frac{3}{2}}}$

Step 4.2.1.1.2

Apply the product rule to $6^{\frac{3}{2}} \sqrt{2}$ .

$t = \pm \frac{{(6^{\frac{3}{2}})}^{\frac{3}{2}} {\sqrt{2}}^{\frac{3}{2}}}{4^{\frac{3}{2}}}$

Step 4.2.1.2

Multiply the exponents in ${(6^{\frac{3}{2}})}^{\frac{3}{2}}$ .

Tap for more steps...

Step 4.2.1.2.1

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

$t = \pm \frac{6^{\frac{3}{2} \cdot \frac{3}{2}} {\sqrt{2}}^{\frac{3}{2}}}{4^{\frac{3}{2}}}$

Step 4.2.1.2.2

Multiply $\frac{3}{2} \cdot \frac{3}{2}$ .

Tap for more steps...

Step 4.2.1.2.2.1

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

$t = \pm \frac{6^{\frac{3 \cdot 3}{2 \cdot 2}} {\sqrt{2}}^{\frac{3}{2}}}{4^{\frac{3}{2}}}$

Step 4.2.1.2.2.2

Multiply $3$ by $3$ .

$t = \pm \frac{6^{\frac{9}{2 \cdot 2}} {\sqrt{2}}^{\frac{3}{2}}}{4^{\frac{3}{2}}}$

Step 4.2.1.2.2.3

Multiply $2$ by $2$ .

$t = \pm \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{4^{\frac{3}{2}}}$

Step 4.2.1.3

Simplify the denominator.

Tap for more steps...

Step 4.2.1.3.1

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

$t = \pm \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{{(2^{2})}^{\frac{3}{2}}}$

Step 4.2.1.3.2

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

$t = \pm \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}$

Step 4.2.1.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.3.3.1

Cancel the common factor.

$t = \pm \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{2^{2 (\frac{3}{2})}}$

Step 4.2.1.3.3.2

Rewrite the expression.

$t = \pm \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{2^{3}}$

Step 4.2.1.3.4

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

$t = \pm \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}$

Step 5

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

Tap for more steps...

Step 5.1

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

$t = \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}$

Step 5.2

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

$t = - \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}$

Step 5.3

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

$t = \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}, - \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}, - \frac{6^{\frac{9}{4}} {\sqrt{2}}^{\frac{3}{2}}}{8}$

Decimal Form:

$t = 11.84466611 \dots, - 11.84466611 \dots$