Precalculus Examples

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

Step 1

Rewrite the equation as $t^{\frac{3}{2}} = \frac{\sqrt{3}}{9^{- 4}}$ .

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

Step 2

Move $9^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 3

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

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

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

Tap for more steps...

Step 4.1.1.1

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

Tap for more steps...

Step 4.1.1.1.1

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

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

Step 4.1.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.1.1.1.2.1

Cancel the common factor.

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

Step 4.1.1.1.2.2

Rewrite the expression.

$t^{\frac{1}{2} \cdot 2} = {(\sqrt{3} \cdot 9^{4})}^{\frac{2}{3}}$

Step 4.1.1.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.1.1.3.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = {(\sqrt{3} \cdot 9^{4})}^{\frac{2}{3}}$

Step 4.1.1.1.3.2

Rewrite the expression.

$t^{1} = {(\sqrt{3} \cdot 9^{4})}^{\frac{2}{3}}$

Step 4.1.1.2

Simplify.

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

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify ${(\sqrt{3} \cdot 9^{4})}^{\frac{2}{3}}$ .

Tap for more steps...

Step 4.2.1.1

Simplify the expression.

Tap for more steps...

Step 4.2.1.1.1

Raise $9$ to the power of $4$ .

$t = {(\sqrt{3} \cdot 6561)}^{\frac{2}{3}}$

Step 4.2.1.1.2

Move $6561$ to the left of $\sqrt{3}$ .

$t = {(6561 \cdot \sqrt{3})}^{\frac{2}{3}}$

Step 4.2.1.1.3

Apply the product rule to $6561 \sqrt{3}$ .

$t = 6561^{\frac{2}{3}} {\sqrt{3}}^{\frac{2}{3}}$

Step 4.2.1.2

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

Tap for more steps...

Step 4.2.1.2.1

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

$t = 6561^{\frac{2}{3}} {(3^{\frac{1}{2}})}^{\frac{2}{3}}$

Step 4.2.1.2.2

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

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{1}{2} \cdot \frac{2}{3}}$

Step 4.2.1.2.3

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{2}{2 \cdot 3}}$

Step 4.2.1.2.4

Multiply $2$ by $3$ .

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{2}{6}}$

Step 4.2.1.2.5

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 4.2.1.2.5.1

Factor $2$ out of $2$ .

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{2 (1)}{6}}$

Step 4.2.1.2.5.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1.2.5.2.1

Factor $2$ out of $6$ .

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{2 \cdot 1}{2 \cdot 3}}$

Step 4.2.1.2.5.2.2

Cancel the common factor.

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{2 \cdot 1}{2 \cdot 3}}$

Step 4.2.1.2.5.2.3

Rewrite the expression.

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{1}{3}}$

Step 4.2.1.2.6

Rewrite $3^{\frac{1}{3}}$ as $\sqrt[3]{3}$ .

$t = 6561^{\frac{2}{3}} \sqrt[3]{3}$

Step 4.2.1.3

Multiply $6561^{\frac{2}{3}} \sqrt[3]{3}$ .

Tap for more steps...

Step 4.2.1.3.1

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

$t = 6561^{\frac{2}{3}} \cdot 3^{\frac{1}{3}}$

Step 4.2.1.3.2

Rewrite $6561$ as $3^{8}$ .

$t = {(3^{8})}^{\frac{2}{3}} \cdot 3^{\frac{1}{3}}$

Step 4.2.1.3.3

Multiply the exponents in ${(3^{8})}^{\frac{2}{3}}$ .

Tap for more steps...

Step 4.2.1.3.3.1

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

$t = 3^{8 (\frac{2}{3})} \cdot 3^{\frac{1}{3}}$

Step 4.2.1.3.3.2

Multiply $8 (\frac{2}{3})$ .

Tap for more steps...

Step 4.2.1.3.3.2.1

Combine $8$ and $\frac{2}{3}$ .

$t = 3^{\frac{8 \cdot 2}{3}} \cdot 3^{\frac{1}{3}}$

Step 4.2.1.3.3.2.2

Multiply $8$ by $2$ .

$t = 3^{\frac{16}{3}} \cdot 3^{\frac{1}{3}}$

Step 4.2.1.3.4

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

$t = 3^{\frac{16}{3} + \frac{1}{3}}$

Step 4.2.1.3.5

Combine the numerators over the common denominator.

$t = 3^{\frac{16 + 1}{3}}$

Step 4.2.1.3.6

Add $16$ and $1$ .

$t = 3^{\frac{17}{3}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$t = 3^{\frac{17}{3}}$

Decimal Form:

$t = 505.46036900 \dots$