Simplify fifth root of (3072h^8)/(243f^5)

$\sqrt[5]{\frac{3072 h^{8}}{243 f^{5}}}$

Step 1

Reduce the expression $\frac{3072 h^{8}}{243 f^{5}}$ by cancelling the common factors.

Factor $3$ out of $3072 h^{8}$ .

$\sqrt[5]{\frac{3 (1024 h^{8})}{243 f^{5}}}$

Factor $3$ out of $243 f^{5}$ .

$\sqrt[5]{\frac{3 (1024 h^{8})}{3 (81 f^{5})}}$

Cancel the common factor.

$\sqrt[5]{\frac{3 (1024 h^{8})}{3 (81 f^{5})}}$

Rewrite the expression.

$\sqrt[5]{\frac{1024 h^{8}}{81 f^{5}}}$

Step 2

Rewrite $\frac{1024 h^{8}}{81 f^{5}}$ as ${(\frac{4 h}{f})}^{5} \frac{h^{3}}{81}$ .

Factor the perfect power ${(4 h)}^{5}$ out of $1024 h^{8}$ .

$\sqrt[5]{\frac{{(4 h)}^{5} h^{3}}{81 f^{5}}}$

Factor the perfect power $f^{5}$ out of $81 f^{5}$ .

$\sqrt[5]{\frac{{(4 h)}^{5} h^{3}}{f^{5} \cdot 81}}$

Rearrange the fraction $\frac{{(4 h)}^{5} h^{3}}{f^{5} \cdot 81}$ .

$\sqrt[5]{{(\frac{4 h}{f})}^{5} \frac{h^{3}}{81}}$

Step 3

Pull terms out from under the radical.

$\frac{4 h}{f} \sqrt[5]{\frac{h^{3}}{81}}$

Step 4

Rewrite $\sqrt[5]{\frac{h^{3}}{81}}$ as $\frac{\sqrt[5]{h^{3}}}{\sqrt[5]{81}}$ .

$\frac{4 h}{f} \cdot \frac{\sqrt[5]{h^{3}}}{\sqrt[5]{81}}$

Step 5

Combine.

$\frac{4 h \sqrt[5]{h^{3}}}{f \sqrt[5]{81}}$

Step 6

Multiply $\frac{4 h \sqrt[5]{h^{3}}}{f \sqrt[5]{81}}$ by $\frac{{\sqrt[5]{81}}^{4}}{{\sqrt[5]{81}}^{4}}$ .

$\frac{4 h \sqrt[5]{h^{3}}}{f \sqrt[5]{81}} \cdot \frac{{\sqrt[5]{81}}^{4}}{{\sqrt[5]{81}}^{4}}$

Step 7

Combine and simplify the denominator.

Multiply $\frac{4 h \sqrt[5]{h^{3}}}{f \sqrt[5]{81}}$ by $\frac{{\sqrt[5]{81}}^{4}}{{\sqrt[5]{81}}^{4}}$ .

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f \sqrt[5]{81} {\sqrt[5]{81}}^{4}}$

Move $\sqrt[5]{81}$ .

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f (\sqrt[5]{81} {\sqrt[5]{81}}^{4})}$

Raise $\sqrt[5]{81}$ to the power of $1$ .

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f ({\sqrt[5]{81}}^{1} {\sqrt[5]{81}}^{4})}$

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

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f {\sqrt[5]{81}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f {\sqrt[5]{81}}^{5}}$

Rewrite ${\sqrt[5]{81}}^{5}$ as $81$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{81}$ as $81^{\frac{1}{5}}$ .

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f {(81^{\frac{1}{5}})}^{5}}$

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

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f \cdot 81^{\frac{1}{5} \cdot 5}}$

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

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f \cdot 81^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f \cdot 81^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f \cdot 81^{1}}$

Evaluate the exponent.

$\frac{4 h \sqrt[5]{h^{3}} {\sqrt[5]{81}}^{4}}{f \cdot 81}$

Step 8

Simplify the numerator.

Rewrite ${\sqrt[5]{81}}^{4}$ as ${(81^{4})}^{\frac{1}{5}}$ .

$\frac{4 h \sqrt[5]{h^{3}} \sqrt[5]{81^{4}}}{f \cdot 81}$

Raise $81$ to the power of $4$ .

$\frac{4 h \sqrt[5]{h^{3}} \sqrt[5]{43046721}}{f \cdot 81}$

Rewrite $43046721$ as $27^{5} \cdot 3$ .

Factor $14348907$ out of $43046721$ .

$\frac{4 h \sqrt[5]{h^{3}} \sqrt[5]{14348907 (3)}}{f \cdot 81}$

Rewrite $14348907$ as $27^{5}$ .

$\frac{4 h \sqrt[5]{h^{3}} \sqrt[5]{27^{5} \cdot 3}}{f \cdot 81}$

Pull terms out from under the radical.

$\frac{4 h \sqrt[5]{h^{3}} \cdot 27 \sqrt[5]{3}}{f \cdot 81}$

Combine exponents.

Multiply $27$ by $4$ .

$\frac{108 h \sqrt[5]{h^{3}} \sqrt[5]{3}}{f \cdot 81}$

Combine using the product rule for radicals.

$\frac{108 h \sqrt[5]{3 h^{3}}}{f \cdot 81}$

Step 9

Reduce the expression by cancelling the common factors.

Cancel the common factor of $108$ and $81$ .

Factor $27$ out of $108 h \sqrt[5]{3 h^{3}}$ .

$\frac{27 (4 h \sqrt[5]{3 h^{3}})}{f \cdot 81}$

Cancel the common factors.

Factor $27$ out of $f \cdot 81$ .

$\frac{27 (4 h \sqrt[5]{3 h^{3}})}{27 (f \cdot 3)}$

Cancel the common factor.

$\frac{27 (4 h \sqrt[5]{3 h^{3}})}{27 (f \cdot 3)}$

Rewrite the expression.

$\frac{4 h \sqrt[5]{3 h^{3}}}{f \cdot 3}$

Move $3$ to the left of $f$ .

$\frac{4 h \sqrt[5]{3 h^{3}}}{3 f}$

Simplify fifth root of (3072h^8)/(243f^5)

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