Simplify (7a)/( fifth root of 4a^7b^13)

$\frac{7 a}{\sqrt[5]{4 a^{7} b^{13}}}$

Step 1

Simplify the denominator.

Rewrite $4 a^{7} b^{13}$ as ${(a b^{2})}^{5} \cdot (4 a^{2} b^{3})$ .

Factor out $a^{5}$ .

$\frac{7 a}{\sqrt[5]{4 (a^{5} a^{2}) b^{13}}}$

Factor out $b^{10}$ .

$\frac{7 a}{\sqrt[5]{4 (a^{5} a^{2}) (b^{10} b^{3})}}$

Rewrite $b^{10}$ as ${(b^{2})}^{5}$ .

$\frac{7 a}{\sqrt[5]{4 (a^{5} a^{2}) ({(b^{2})}^{5} b^{3})}}$

Move $a^{2}$ .

$\frac{7 a}{\sqrt[5]{4 (a^{5}) {(b^{2})}^{5} a^{2} b^{3}}}$

Move $4$ .

$\frac{7 a}{\sqrt[5]{a^{5} {(b^{2})}^{5} \cdot 4 a^{2} b^{3}}}$

Rewrite $(a^{5}) {(b^{2})}^{5}$ as ${(a b^{2})}^{5}$ .

$\frac{7 a}{\sqrt[5]{{(a b^{2})}^{5} \cdot 4 a^{2} b^{3}}}$

Add parentheses.

$\frac{7 a}{\sqrt[5]{{(a b^{2})}^{5} \cdot 4 (a^{2} b^{3})}}$

Add parentheses.

$\frac{7 a}{\sqrt[5]{{(a b^{2})}^{5} \cdot (4 a^{2} b^{3})}}$

Pull terms out from under the radical.

$\frac{7 a}{a b^{2} \sqrt[5]{4 a^{2} b^{3}}}$

Step 2

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{7 a}{a b^{2} \sqrt[5]{4 a^{2} b^{3}}}$

Rewrite the expression.

$\frac{7}{b^{2} \sqrt[5]{4 a^{2} b^{3}}}$

Step 3

Multiply $\frac{7}{b^{2} \sqrt[5]{4 a^{2} b^{3}}}$ by $\frac{{\sqrt[5]{4 a^{2} b^{3}}}^{4}}{{\sqrt[5]{4 a^{2} b^{3}}}^{4}}$ .

$\frac{7}{b^{2} \sqrt[5]{4 a^{2} b^{3}}} \cdot \frac{{\sqrt[5]{4 a^{2} b^{3}}}^{4}}{{\sqrt[5]{4 a^{2} b^{3}}}^{4}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{7}{b^{2} \sqrt[5]{4 a^{2} b^{3}}}$ by $\frac{{\sqrt[5]{4 a^{2} b^{3}}}^{4}}{{\sqrt[5]{4 a^{2} b^{3}}}^{4}}$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} \sqrt[5]{4 a^{2} b^{3}} {\sqrt[5]{4 a^{2} b^{3}}}^{4}}$

Move $\sqrt[5]{4 a^{2} b^{3}}$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} (\sqrt[5]{4 a^{2} b^{3}} {\sqrt[5]{4 a^{2} b^{3}}}^{4})}$

Raise $\sqrt[5]{4 a^{2} b^{3}}$ to the power of $1$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} ({\sqrt[5]{4 a^{2} b^{3}}}^{1} {\sqrt[5]{4 a^{2} b^{3}}}^{4})}$

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

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {\sqrt[5]{4 a^{2} b^{3}}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {\sqrt[5]{4 a^{2} b^{3}}}^{5}}$

Rewrite ${\sqrt[5]{4 a^{2} b^{3}}}^{5}$ as $4 a^{2} b^{3}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4 a^{2} b^{3}}$ as ${(4 a^{2} b^{3})}^{\frac{1}{5}}$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {({(4 a^{2} b^{3})}^{\frac{1}{5}})}^{5}}$

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

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {(4 a^{2} b^{3})}^{\frac{1}{5} \cdot 5}}$

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

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {(4 a^{2} b^{3})}^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {(4 a^{2} b^{3})}^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} {(4 a^{2} b^{3})}^{1}}$

Simplify.

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{2} (4 a^{2} b^{3})}$

Step 5

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

Move $b^{3}$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{3} b^{2} (4 a^{2})}$

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

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{3 + 2} (4 a^{2})}$

Add $3$ and $2$ .

$\frac{7 {\sqrt[5]{4 a^{2} b^{3}}}^{4}}{b^{5} (4 a^{2})}$

Step 6

Simplify the numerator.

Rewrite ${\sqrt[5]{4 a^{2} b^{3}}}^{4}$ as $\sqrt[5]{{(4 a^{2} b^{3})}^{4}}$ .

$\frac{7 \sqrt[5]{{(4 a^{2} b^{3})}^{4}}}{b^{5} \cdot 4 a^{2}}$

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

$\frac{7 \sqrt[5]{{(4 a^{2})}^{4} {(b^{3})}^{4}}}{b^{5} \cdot 4 a^{2}}$

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

$\frac{7 \sqrt[5]{4^{4} {(a^{2})}^{4} {(b^{3})}^{4}}}{b^{5} \cdot 4 a^{2}}$

Raise $4$ to the power of $4$ .

$\frac{7 \sqrt[5]{256 {(a^{2})}^{4} {(b^{3})}^{4}}}{b^{5} \cdot 4 a^{2}}$

Multiply the exponents in ${(a^{2})}^{4}$ .

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

$\frac{7 \sqrt[5]{256 a^{2 \cdot 4} {(b^{3})}^{4}}}{b^{5} \cdot 4 a^{2}}$

Multiply $2$ by $4$ .

$\frac{7 \sqrt[5]{256 a^{8} {(b^{3})}^{4}}}{b^{5} \cdot 4 a^{2}}$

Multiply the exponents in ${(b^{3})}^{4}$ .

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

$\frac{7 \sqrt[5]{256 a^{8} b^{3 \cdot 4}}}{b^{5} \cdot 4 a^{2}}$

Multiply $3$ by $4$ .

$\frac{7 \sqrt[5]{256 a^{8} b^{12}}}{b^{5} \cdot 4 a^{2}}$

Rewrite $256 a^{8} b^{12}$ as ${(2 a b^{2})}^{5} \cdot (8 a^{3} b^{2})$ .

Factor $32$ out of $256$ .

$\frac{7 \sqrt[5]{32 (8) a^{8} b^{12}}}{b^{5} \cdot 4 a^{2}}$

Rewrite $32$ as $2^{5}$ .

$\frac{7 \sqrt[5]{2^{5} \cdot 8 a^{8} b^{12}}}{b^{5} \cdot 4 a^{2}}$

Factor out $a^{5}$ .

$\frac{7 \sqrt[5]{2^{5} \cdot 8 (a^{5} a^{3}) b^{12}}}{b^{5} \cdot 4 a^{2}}$

Factor out $b^{10}$ .

$\frac{7 \sqrt[5]{2^{5} \cdot 8 (a^{5} a^{3}) (b^{10} b^{2})}}{b^{5} \cdot 4 a^{2}}$

Rewrite $b^{10}$ as ${(b^{2})}^{5}$ .

$\frac{7 \sqrt[5]{2^{5} \cdot 8 (a^{5} a^{3}) ({(b^{2})}^{5} b^{2})}}{b^{5} \cdot 4 a^{2}}$

Move $a^{3}$ .

$\frac{7 \sqrt[5]{2^{5} \cdot 8 (a^{5}) {(b^{2})}^{5} a^{3} b^{2}}}{b^{5} \cdot 4 a^{2}}$

Move $8$ .

$\frac{7 \sqrt[5]{2^{5} (a^{5}) {(b^{2})}^{5} \cdot 8 a^{3} b^{2}}}{b^{5} \cdot 4 a^{2}}$

Rewrite $(2^{5} (a^{5})) {(b^{2})}^{5}$ as ${(2 a b^{2})}^{5}$ .

$\frac{7 \sqrt[5]{{(2 a b^{2})}^{5} \cdot 8 a^{3} b^{2}}}{b^{5} \cdot 4 a^{2}}$

Add parentheses.

$\frac{7 \sqrt[5]{{(2 a b^{2})}^{5} \cdot 8 (a^{3} b^{2})}}{b^{5} \cdot 4 a^{2}}$

Add parentheses.

$\frac{7 \sqrt[5]{{(2 a b^{2})}^{5} \cdot (8 a^{3} b^{2})}}{b^{5} \cdot 4 a^{2}}$

Pull terms out from under the radical.

$\frac{7 \cdot 2 a b^{2} \sqrt[5]{8 a^{3} b^{2}}}{b^{5} \cdot 4 a^{2}}$

Multiply $7$ by $2$ .

$\frac{14 a b^{2} \sqrt[5]{8 a^{3} b^{2}}}{b^{5} \cdot 4 a^{2}}$

Step 7

Reduce the expression by cancelling the common factors.

Cancel the common factor of $14$ and $4$ .

Factor $2$ out of $14 a b^{2} \sqrt[5]{8 a^{3} b^{2}}$ .

$\frac{2 (7 a b^{2} \sqrt[5]{8 a^{3} b^{2}})}{b^{5} \cdot 4 a^{2}}$

Cancel the common factors.

Factor $2$ out of $b^{5} \cdot 4 a^{2}$ .

$\frac{2 (7 a b^{2} \sqrt[5]{8 a^{3} b^{2}})}{2 (b^{5} \cdot 2 a^{2})}$

Cancel the common factor.

$\frac{2 (7 a b^{2} \sqrt[5]{8 a^{3} b^{2}})}{2 (b^{5} \cdot 2 a^{2})}$

Rewrite the expression.

$\frac{7 a b^{2} \sqrt[5]{8 a^{3} b^{2}}}{b^{5} \cdot 2 a^{2}}$

Cancel the common factor of $a$ and $a^{2}$ .

Factor $a$ out of $7 a b^{2} \sqrt[5]{8 a^{3} b^{2}}$ .

$\frac{a (7 b^{2} \sqrt[5]{8 a^{3} b^{2}})}{b^{5} \cdot 2 a^{2}}$

Cancel the common factors.

Factor $a$ out of $b^{5} \cdot 2 a^{2}$ .

$\frac{a (7 b^{2} \sqrt[5]{8 a^{3} b^{2}})}{a (b^{5} \cdot 2 a)}$

Cancel the common factor.

$\frac{a (7 b^{2} \sqrt[5]{8 a^{3} b^{2}})}{a (b^{5} \cdot 2 a)}$

Rewrite the expression.

$\frac{7 b^{2} \sqrt[5]{8 a^{3} b^{2}}}{b^{5} \cdot 2 a}$

Cancel the common factor of $b^{2}$ and $b^{5}$ .

Factor $b^{2}$ out of $7 b^{2} \sqrt[5]{8 a^{3} b^{2}}$ .

$\frac{b^{2} (7 \sqrt[5]{8 a^{3} b^{2}})}{b^{5} \cdot 2 a}$

Cancel the common factors.

Factor $b^{2}$ out of $b^{5} \cdot 2 a$ .

$\frac{b^{2} (7 \sqrt[5]{8 a^{3} b^{2}})}{b^{2} (b^{3} \cdot 2 a)}$

Cancel the common factor.

$\frac{b^{2} (7 \sqrt[5]{8 a^{3} b^{2}})}{b^{2} (b^{3} \cdot 2 a)}$

Rewrite the expression.

$\frac{7 \sqrt[5]{8 a^{3} b^{2}}}{b^{3} \cdot 2 a}$

Move $2$ to the left of $b^{3}$ .

$\frac{7 \sqrt[5]{8 a^{3} b^{2}}}{2 b^{3} a}$

Simplify (7a)/( fifth root of 4a^7b^13)

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