Simplify the Radical Expression (18w^(1/3)v^(5/4))/(27w^(4/3)v^(1/2))

$\frac{18 w^{\frac{1}{3}} v^{\frac{5}{4}}}{27 w^{\frac{4}{3}} v^{\frac{1}{2}}}$

Step 1

Move $w^{\frac{1}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{18 v^{\frac{5}{4}}}{27 w^{\frac{4}{3}} v^{\frac{1}{2}} w^{- \frac{1}{3}}}$

Step 2

Simplify the denominator.

Multiply $w^{\frac{4}{3}}$ by $w^{- \frac{1}{3}}$ by adding the exponents.

Move $w^{- \frac{1}{3}}$ .

$\frac{18 v^{\frac{5}{4}}}{27 (w^{- \frac{1}{3}} w^{\frac{4}{3}}) v^{\frac{1}{2}}}$

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

$\frac{18 v^{\frac{5}{4}}}{27 w^{- \frac{1}{3} + \frac{4}{3}} v^{\frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{18 v^{\frac{5}{4}}}{27 w^{\frac{- 1 + 4}{3}} v^{\frac{1}{2}}}$

Add $- 1$ and $4$ .

$\frac{18 v^{\frac{5}{4}}}{27 w^{\frac{3}{3}} v^{\frac{1}{2}}}$

Divide $3$ by $3$ .

$\frac{18 v^{\frac{5}{4}}}{27 w^{1} v^{\frac{1}{2}}}$

Simplify $27 w^{1} v^{\frac{1}{2}}$ .

$\frac{18 v^{\frac{5}{4}}}{27 w v^{\frac{1}{2}}}$

Step 3

Move $v^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{18 v^{\frac{5}{4}} v^{- \frac{1}{2}}}{27 w}$

Step 4

Multiply $v^{\frac{5}{4}}$ by $v^{- \frac{1}{2}}$ by adding the exponents.

Move $v^{- \frac{1}{2}}$ .

$\frac{18 (v^{- \frac{1}{2}} v^{\frac{5}{4}})}{27 w}$

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

$\frac{18 v^{- \frac{1}{2} + \frac{5}{4}}}{27 w}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{18 v^{- \frac{1}{2} \cdot \frac{2}{2} + \frac{5}{4}}}{27 w}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{18 v^{- \frac{2}{2 \cdot 2} + \frac{5}{4}}}{27 w}$

Multiply $2$ by $2$ .

$\frac{18 v^{- \frac{2}{4} + \frac{5}{4}}}{27 w}$

Combine the numerators over the common denominator.

$\frac{18 v^{\frac{- 2 + 5}{4}}}{27 w}$

Add $- 2$ and $5$ .

$\frac{18 v^{\frac{3}{4}}}{27 w}$

Step 5

Factor $9$ out of $18 v^{\frac{3}{4}}$ .

$\frac{9 (2 v^{\frac{3}{4}})}{27 w}$

Step 6

Cancel the common factors.

Factor $9$ out of $27 w$ .

$\frac{9 (2 v^{\frac{3}{4}})}{9 (3 w)}$

Cancel the common factor.

$\frac{9 (2 v^{\frac{3}{4}})}{9 (3 w)}$

Rewrite the expression.

$\frac{2 v^{\frac{3}{4}}}{3 w}$

Simplify the Radical Expression (18w^(1/3)v^(5/4))/(27w^(4/3)v^(1/2))

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