Simplify (3ab^(1/2))/(9a^(1/2)b^2)

$\frac{3 a b^{\frac{1}{2}}}{9 a^{\frac{1}{2}} b^{2}}$

Step 1

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

$\frac{3 a}{9 a^{\frac{1}{2}} b^{2} b^{- \frac{1}{2}}}$

Step 2

Multiply $b^{2}$ by $b^{- \frac{1}{2}}$ by adding the exponents.

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

$\frac{3 a}{9 a^{\frac{1}{2}} (b^{- \frac{1}{2}} b^{2})}$

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

$\frac{3 a}{9 a^{\frac{1}{2}} b^{- \frac{1}{2} + 2}}$

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

$\frac{3 a}{9 a^{\frac{1}{2}} b^{- \frac{1}{2} + 2 \cdot \frac{2}{2}}}$

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

$\frac{3 a}{9 a^{\frac{1}{2}} b^{- \frac{1}{2} + \frac{2 \cdot 2}{2}}}$

Combine the numerators over the common denominator.

$\frac{3 a}{9 a^{\frac{1}{2}} b^{\frac{- 1 + 2 \cdot 2}{2}}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{3 a}{9 a^{\frac{1}{2}} b^{\frac{- 1 + 4}{2}}}$

Add $- 1$ and $4$ .

$\frac{3 a}{9 a^{\frac{1}{2}} b^{\frac{3}{2}}}$

Step 3

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

$\frac{3 a \cdot a^{- \frac{1}{2}}}{9 b^{\frac{3}{2}}}$

Step 4

Multiply $a$ by $a^{- \frac{1}{2}}$ by adding the exponents.

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

$\frac{3 (a^{- \frac{1}{2}} a)}{9 b^{\frac{3}{2}}}$

Multiply $a^{- \frac{1}{2}}$ by $a$ .

Raise $a$ to the power of $1$ .

$\frac{3 (a^{- \frac{1}{2}} a^{1})}{9 b^{\frac{3}{2}}}$

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

$\frac{3 a^{- \frac{1}{2} + 1}}{9 b^{\frac{3}{2}}}$

Write $1$ as a fraction with a common denominator.

$\frac{3 a^{- \frac{1}{2} + \frac{2}{2}}}{9 b^{\frac{3}{2}}}$

Combine the numerators over the common denominator.

$\frac{3 a^{\frac{- 1 + 2}{2}}}{9 b^{\frac{3}{2}}}$

Add $- 1$ and $2$ .

$\frac{3 a^{\frac{1}{2}}}{9 b^{\frac{3}{2}}}$

Step 5

Factor $3$ out of $3 a^{\frac{1}{2}}$ .

$\frac{3 (a^{\frac{1}{2}})}{9 b^{\frac{3}{2}}}$

Step 6

Cancel the common factors.

Factor $3$ out of $9 b^{\frac{3}{2}}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (3ab^(1/2))/(9a^(1/2)b^2)

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