Simplify (16a^(1/8)b^-6c^(3/5))/(2^3a^(3/8)b^(-1/5)c^(2/5))

$\frac{16 a^{\frac{1}{8}} b^{- 6} c^{\frac{3}{5}}}{2^{3} a^{\frac{3}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}}}$

Step 1

Move $a^{\frac{1}{8}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{16 b^{- 6} c^{\frac{3}{5}}}{2^{3} a^{\frac{3}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}} a^{- \frac{1}{8}}}$

Step 2

Move $b^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{3}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}} a^{- \frac{1}{8}} b^{6}}$

Step 3

Simplify the denominator.

Multiply $a^{\frac{3}{8}}$ by $a^{- \frac{1}{8}}$ by adding the exponents.

Move $a^{- \frac{1}{8}}$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} (a^{- \frac{1}{8}} a^{\frac{3}{8}}) b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

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

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{- \frac{1}{8} + \frac{3}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Combine the numerators over the common denominator.

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{- 1 + 3}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Add $- 1$ and $3$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{2}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{2 (1)}{8}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{2 \cdot 1}{2 \cdot 4}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Cancel the common factor.

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{2 \cdot 1}{2 \cdot 4}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Rewrite the expression.

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{- \frac{1}{5}} c^{\frac{2}{5}} b^{6}}$

Multiply $b^{- \frac{1}{5}}$ by $b^{6}$ by adding the exponents.

Move $b^{6}$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} (b^{6} b^{- \frac{1}{5}}) c^{\frac{2}{5}}}$

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

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{6 - \frac{1}{5}} c^{\frac{2}{5}}}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{6 \cdot \frac{5}{5} - \frac{1}{5}} c^{\frac{2}{5}}}$

Combine $6$ and $\frac{5}{5}$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{6 \cdot 5}{5} - \frac{1}{5}} c^{\frac{2}{5}}}$

Combine the numerators over the common denominator.

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{6 \cdot 5 - 1}{5}} c^{\frac{2}{5}}}$

Simplify the numerator.

Multiply $6$ by $5$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{30 - 1}{5}} c^{\frac{2}{5}}}$

Subtract $1$ from $30$ .

$\frac{16 c^{\frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{29}{5}} c^{\frac{2}{5}}}$

Step 4

Move $c^{\frac{2}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{16 c^{\frac{3}{5}} c^{- \frac{2}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Step 5

Multiply $c^{\frac{3}{5}}$ by $c^{- \frac{2}{5}}$ by adding the exponents.

Move $c^{- \frac{2}{5}}$ .

$\frac{16 (c^{- \frac{2}{5}} c^{\frac{3}{5}})}{2^{3} a^{\frac{1}{4}} b^{\frac{29}{5}}}$

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

$\frac{16 c^{- \frac{2}{5} + \frac{3}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Combine the numerators over the common denominator.

$\frac{16 c^{\frac{- 2 + 3}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Add $- 2$ and $3$ .

$\frac{16 c^{\frac{1}{5}}}{2^{3} a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Step 6

Raise $2$ to the power of $3$ .

$\frac{16 c^{\frac{1}{5}}}{8 a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Step 7

Factor $8$ out of $16 c^{\frac{1}{5}}$ .

$\frac{8 (2 c^{\frac{1}{5}})}{8 a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Step 8

Cancel the common factors.

Factor $8$ out of $8 a^{\frac{1}{4}} b^{\frac{29}{5}}$ .

$\frac{8 (2 c^{\frac{1}{5}})}{8 (a^{\frac{1}{4}} b^{\frac{29}{5}})}$

Cancel the common factor.

$\frac{8 (2 c^{\frac{1}{5}})}{8 (a^{\frac{1}{4}} b^{\frac{29}{5}})}$

Rewrite the expression.

$\frac{2 c^{\frac{1}{5}}}{a^{\frac{1}{4}} b^{\frac{29}{5}}}$

Simplify (16a^(1/8)b^-6c^(3/5))/(2^3a^(3/8)b^(-1/5)c^(2/5))

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