Algebra Examples

${(t v^{\frac{1}{3}})}^{2} {(t^{2} v^{- 3})}^{- \frac{1}{2}}$

Step 1

Apply the product rule to $t v^{\frac{1}{3}}$ .

$t^{2} {(v^{\frac{1}{3}})}^{2} {(t^{2} v^{- 3})}^{- \frac{1}{2}}$

Step 2

Multiply the exponents in ${(v^{\frac{1}{3}})}^{2}$ .

Tap for more steps...

Step 2.1

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

$t^{2} v^{\frac{1}{3} \cdot 2} {(t^{2} v^{- 3})}^{- \frac{1}{2}}$

Step 2.2

Combine $\frac{1}{3}$ and $2$ .

$t^{2} v^{\frac{2}{3}} {(t^{2} v^{- 3})}^{- \frac{1}{2}}$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$t^{2} v^{\frac{2}{3}} {(t^{2} \frac{1}{v^{3}})}^{- \frac{1}{2}}$

Step 4

Combine $t^{2}$ and $\frac{1}{v^{3}}$ .

$t^{2} v^{\frac{2}{3}} {(\frac{t^{2}}{v^{3}})}^{- \frac{1}{2}}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 6

Apply basic rules of exponents.

Tap for more steps...

Step 6.1

Apply the product rule to $\frac{v^{3}}{t^{2}}$ .

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

Step 6.2

Multiply the exponents in ${(v^{3})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 7

Simplify the denominator.

Tap for more steps...

Step 7.1

Multiply the exponents in ${(t^{2})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.1.2.1

Cancel the common factor.

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

Step 7.1.2.2

Rewrite the expression.

$t^{2} v^{\frac{2}{3}} \frac{v^{\frac{3}{2}}}{t^{1}}$

Step 7.2

Simplify.

$t^{2} v^{\frac{2}{3}} \frac{v^{\frac{3}{2}}}{t}$

Step 8

Cancel the common factor of $t$ .

Tap for more steps...

Step 8.1

Factor $t$ out of $t^{2} v^{\frac{2}{3}}$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

$t v^{\frac{2}{3}} v^{\frac{3}{2}}$

Step 9

Multiply $v^{\frac{2}{3}}$ by $v^{\frac{3}{2}}$ by adding the exponents.

Tap for more steps...

Step 9.1

Move $v^{\frac{3}{2}}$ .

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

Step 9.2

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

$t v^{\frac{3}{2} + \frac{2}{3}}$

Step 9.3

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$t v^{\frac{3}{2} \cdot \frac{3}{3} + \frac{2}{3}}$

Step 9.4

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$t v^{\frac{3}{2} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 9.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.5.1

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

$t v^{\frac{3 \cdot 3}{2 \cdot 3} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 9.5.2

Multiply $2$ by $3$ .

$t v^{\frac{3 \cdot 3}{6} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 9.5.3

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

$t v^{\frac{3 \cdot 3}{6} + \frac{2 \cdot 2}{3 \cdot 2}}$

Step 9.5.4

Multiply $3$ by $2$ .

$t v^{\frac{3 \cdot 3}{6} + \frac{2 \cdot 2}{6}}$

Step 9.6

Combine the numerators over the common denominator.

$t v^{\frac{3 \cdot 3 + 2 \cdot 2}{6}}$

Step 9.7

Simplify the numerator.

Tap for more steps...

Step 9.7.1

Multiply $3$ by $3$ .

$t v^{\frac{9 + 2 \cdot 2}{6}}$

Step 9.7.2

Multiply $2$ by $2$ .

$t v^{\frac{9 + 4}{6}}$

Step 9.7.3

Add $9$ and $4$ .

$t v^{\frac{13}{6}}$