Simplify (6a^2b^(3/2))/(12a^(3/2)b^3)

$\frac{6 a^{2} b^{\frac{3}{2}}}{12 a^{\frac{3}{2}} b^{3}}$

Move $b^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{3} b^{- \frac{3}{2}}}$

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

Move $b^{- \frac{3}{2}}$ .

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} (b^{- \frac{3}{2}} b^{3})}$

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

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{- \frac{3}{2} + 3}}$

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

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{- \frac{3}{2} + 3 \cdot \frac{2}{2}}}$

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

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{- \frac{3}{2} + \frac{3 \cdot 2}{2}}}$

Combine the numerators over the common denominator.

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{\frac{- 3 + 3 \cdot 2}{2}}}$

Simplify the numerator.

Multiply $3$ by $2$ .

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{\frac{- 3 + 6}{2}}}$

Add $- 3$ and $6$ .

$\frac{6 a^{2}}{12 a^{\frac{3}{2}} b^{\frac{3}{2}}}$

Move $a^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{6 a^{2} a^{- \frac{3}{2}}}{12 b^{\frac{3}{2}}}$

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

Move $a^{- \frac{3}{2}}$ .

$\frac{6 (a^{- \frac{3}{2}} a^{2})}{12 b^{\frac{3}{2}}}$

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

$\frac{6 a^{- \frac{3}{2} + 2}}{12 b^{\frac{3}{2}}}$

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

$\frac{6 a^{- \frac{3}{2} + 2 \cdot \frac{2}{2}}}{12 b^{\frac{3}{2}}}$

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

$\frac{6 a^{- \frac{3}{2} + \frac{2 \cdot 2}{2}}}{12 b^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{6 a^{\frac{- 3 + 2 \cdot 2}{2}}}{12 b^{\frac{3}{2}}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{6 a^{\frac{- 3 + 4}{2}}}{12 b^{\frac{3}{2}}}$

Add $- 3$ and $4$ .

$\frac{6 a^{\frac{1}{2}}}{12 b^{\frac{3}{2}}}$

Factor $6$ out of $6 a^{\frac{1}{2}}$ .

$\frac{6 (a^{\frac{1}{2}})}{12 b^{\frac{3}{2}}}$

Cancel the common factors.

Factor $6$ out of $12 b^{\frac{3}{2}}$ .

$\frac{6 (a^{\frac{1}{2}})}{6 (2 b^{\frac{3}{2}})}$

Cancel the common factor.

$\frac{6 a^{\frac{1}{2}}}{6 (2 b^{\frac{3}{2}})}$

Rewrite the expression.

$\frac{a^{\frac{1}{2}}}{2 b^{\frac{3}{2}}}$

Simplify (6a^2b^(3/2))/(12a^(3/2)b^3)

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