Simplify (2^3a^2b^4c^3)^(1/5)(2a^4b^3c^5)^(1/6)

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

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

${(8 a^{2} b^{4} c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $8 a^{2} b^{4} c^{3}$ .

${(8 a^{2} b^{4})}^{\frac{1}{5}} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Apply the product rule to $8 a^{2} b^{4}$ .

${(8 a^{2})}^{\frac{1}{5}} {(b^{4})}^{\frac{1}{5}} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Apply the product rule to $8 a^{2}$ .

$8^{\frac{1}{5}} {(a^{2})}^{\frac{1}{5}} {(b^{4})}^{\frac{1}{5}} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Multiply the exponents in ${(a^{2})}^{\frac{1}{5}}$ .

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

$8^{\frac{1}{5}} a^{2 (\frac{1}{5})} {(b^{4})}^{\frac{1}{5}} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Combine $2$ and $\frac{1}{5}$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} {(b^{4})}^{\frac{1}{5}} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Multiply the exponents in ${(b^{4})}^{\frac{1}{5}}$ .

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

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{4 (\frac{1}{5})} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Combine $4$ and $\frac{1}{5}$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} {(c^{3})}^{\frac{1}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Multiply the exponents in ${(c^{3})}^{\frac{1}{5}}$ .

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

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{3 (\frac{1}{5})} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Combine $3$ and $\frac{1}{5}$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} {(2 a^{4} b^{3} c^{5})}^{\frac{1}{6}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 a^{4} b^{3} c^{5}$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} ({(2 a^{4} b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Apply the product rule to $2 a^{4} b^{3}$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} ({(2 a^{4})}^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Apply the product rule to $2 a^{4}$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(a^{4})}^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply the exponents in ${(a^{4})}^{\frac{1}{6}}$ .

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

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} a^{4 (\frac{1}{6})} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} a^{2 (2) \frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Factor $2$ out of $6$ .

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} a^{2 \cdot 2 \frac{1}{2 \cdot 3}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Cancel the common factor.

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} a^{2 \cdot 2 \frac{1}{2 \cdot 3}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Rewrite the expression.

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} a^{2 (\frac{1}{3})} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

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

$8^{\frac{1}{5}} a^{\frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} a^{\frac{2}{3}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

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

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

$8^{\frac{1}{5}} (a^{\frac{2}{3}} a^{\frac{2}{5}}) b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

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

$8^{\frac{1}{5}} a^{\frac{2}{3} + \frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$8^{\frac{1}{5}} a^{\frac{2}{3} \cdot \frac{5}{5} + \frac{2}{5}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

To write $\frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8^{\frac{1}{5}} a^{\frac{2}{3} \cdot \frac{5}{5} + \frac{2}{5} \cdot \frac{3}{3}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$8^{\frac{1}{5}} a^{\frac{2 \cdot 5}{3 \cdot 5} + \frac{2}{5} \cdot \frac{3}{3}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $3$ by $5$ .

$8^{\frac{1}{5}} a^{\frac{2 \cdot 5}{15} + \frac{2}{5} \cdot \frac{3}{3}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $\frac{2}{5}$ by $\frac{3}{3}$ .

$8^{\frac{1}{5}} a^{\frac{2 \cdot 5}{15} + \frac{2 \cdot 3}{5 \cdot 3}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $5$ by $3$ .

$8^{\frac{1}{5}} a^{\frac{2 \cdot 5}{15} + \frac{2 \cdot 3}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Combine the numerators over the common denominator.

$8^{\frac{1}{5}} a^{\frac{2 \cdot 5 + 2 \cdot 3}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Simplify the numerator.

Multiply $2$ by $5$ .

$8^{\frac{1}{5}} a^{\frac{10 + 2 \cdot 3}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $2$ by $3$ .

$8^{\frac{1}{5}} a^{\frac{10 + 6}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Add $10$ and $6$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(b^{3})}^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply the exponents in ${(b^{3})}^{\frac{1}{6}}$ .

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

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} b^{3 (\frac{1}{6})} {(c^{5})}^{\frac{1}{6}})$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} b^{3 \frac{1}{3 (2)}} {(c^{5})}^{\frac{1}{6}})$

Cancel the common factor.

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} b^{3 \frac{1}{3 \cdot 2}} {(c^{5})}^{\frac{1}{6}})$

Rewrite the expression.

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} b^{\frac{1}{2}} {(c^{5})}^{\frac{1}{6}})$

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

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

$8^{\frac{1}{5}} a^{\frac{16}{15}} (b^{\frac{1}{2}} b^{\frac{4}{5}}) c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

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

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{1}{2} + \frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{5}{2 \cdot 5} + \frac{4}{5} \cdot \frac{2}{2}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $2$ by $5$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{5}{10} + \frac{4}{5} \cdot \frac{2}{2}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $\frac{4}{5}$ by $\frac{2}{2}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{5}{10} + \frac{4 \cdot 2}{5 \cdot 2}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply $5$ by $2$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{5}{10} + \frac{4 \cdot 2}{10}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Combine the numerators over the common denominator.

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{5 + 4 \cdot 2}{10}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Simplify the numerator.

Multiply $4$ by $2$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{5 + 8}{10}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Add $5$ and $8$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{3}{5}} (2^{\frac{1}{6}} {(c^{5})}^{\frac{1}{6}})$

Multiply the exponents in ${(c^{5})}^{\frac{1}{6}}$ .

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

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{3}{5}} (2^{\frac{1}{6}} c^{5 (\frac{1}{6})})$

Combine $5$ and $\frac{1}{6}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{3}{5}} (2^{\frac{1}{6}} c^{\frac{5}{6}})$

Multiply $c^{\frac{3}{5}}$ by $c^{\frac{5}{6}}$ by adding the exponents.

Move $c^{\frac{5}{6}}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} (c^{\frac{5}{6}} c^{\frac{3}{5}}) \cdot 2^{\frac{1}{6}}$

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

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5}{6} + \frac{3}{5}} \cdot 2^{\frac{1}{6}}$

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

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5}{6} \cdot \frac{5}{5} + \frac{3}{5}} \cdot 2^{\frac{1}{6}}$

To write $\frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5}{6} \cdot \frac{5}{5} + \frac{3}{5} \cdot \frac{6}{6}} \cdot 2^{\frac{1}{6}}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{6}$ by $\frac{5}{5}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5 \cdot 5}{6 \cdot 5} + \frac{3}{5} \cdot \frac{6}{6}} \cdot 2^{\frac{1}{6}}$

Multiply $6$ by $5$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5 \cdot 5}{30} + \frac{3}{5} \cdot \frac{6}{6}} \cdot 2^{\frac{1}{6}}$

Multiply $\frac{3}{5}$ by $\frac{6}{6}$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5 \cdot 5}{30} + \frac{3 \cdot 6}{5 \cdot 6}} \cdot 2^{\frac{1}{6}}$

Multiply $5$ by $6$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5 \cdot 5}{30} + \frac{3 \cdot 6}{30}} \cdot 2^{\frac{1}{6}}$

Combine the numerators over the common denominator.

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{5 \cdot 5 + 3 \cdot 6}{30}} \cdot 2^{\frac{1}{6}}$

Simplify the numerator.

Multiply $5$ by $5$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{25 + 3 \cdot 6}{30}} \cdot 2^{\frac{1}{6}}$

Multiply $3$ by $6$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{25 + 18}{30}} \cdot 2^{\frac{1}{6}}$

Add $25$ and $18$ .

$8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}} \cdot 2^{\frac{1}{6}}$

Multiply $8^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}} \cdot 2^{\frac{1}{6}}$ .

Rewrite $8$ as $2^{3}$ .

$2^{\frac{1}{6}} \cdot {(2^{3})}^{\frac{1}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Multiply the exponents in ${(2^{3})}^{\frac{1}{5}}$ .

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

$2^{\frac{1}{6}} \cdot 2^{3 (\frac{1}{5})} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Combine $3$ and $\frac{1}{5}$ .

$2^{\frac{1}{6}} \cdot 2^{\frac{3}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

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

$2^{\frac{1}{6} + \frac{3}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2^{\frac{1}{6} \cdot \frac{5}{5} + \frac{3}{5}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

To write $\frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2^{\frac{1}{6} \cdot \frac{5}{5} + \frac{3}{5} \cdot \frac{6}{6}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{6}$ by $\frac{5}{5}$ .

$2^{\frac{5}{6 \cdot 5} + \frac{3}{5} \cdot \frac{6}{6}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Multiply $6$ by $5$ .

$2^{\frac{5}{30} + \frac{3}{5} \cdot \frac{6}{6}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Multiply $\frac{3}{5}$ by $\frac{6}{6}$ .

$2^{\frac{5}{30} + \frac{3 \cdot 6}{5 \cdot 6}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Multiply $5$ by $6$ .

$2^{\frac{5}{30} + \frac{3 \cdot 6}{30}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Combine the numerators over the common denominator.

$2^{\frac{5 + 3 \cdot 6}{30}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Simplify the numerator.

Multiply $3$ by $6$ .

$2^{\frac{5 + 18}{30}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Add $5$ and $18$ .

$2^{\frac{23}{30}} a^{\frac{16}{15}} b^{\frac{13}{10}} c^{\frac{43}{30}}$

Simplify (2^3a^2b^4c^3)^(1/5)(2a^4b^3c^5)^(1/6)

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