Solve (frac{5a^{1/3}b^{3}}{4a^{2}b^{5}})^-4div(a^2/5b^3)^-2

$$( \frac { 5 a ^ { 1 / 3 } b ^ { 3 } } { 4 a ^ { 2 } b ^ { 5 } } ) ^ { - 4 } \div ( \frac { a ^ { 2 } } { 5 b ^ { 3 } } ) ^ { - 2 }$$

Evaluate

$$\frac{256b^{2}a^{\frac{32}{3}}}{15625}$$

Solution Steps

Cancel out \sqrt[3]{a}b^{3} in both numerator and denominator.

$$\frac{\left(\frac{5}{4a^{\frac{5}{3}}b^{2}}\right)^{-4}}{\left(\frac{a^{2}}{5b^{3}}\right)^{-2}}$$

To raise \frac{5}{4a^{\frac{5}{3}}b^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}}}{\left(\frac{a^{2}}{5b^{3}}\right)^{-2}}$$

To raise \frac{a^{2}}{5b^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}}}{\frac{\left(a^{2}\right)^{-2}}{\left(5b^{3}\right)^{-2}}}$$

Divide \frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}} by \frac{\left(a^{2}\right)^{-2}}{\left(5b^{3}\right)^{-2}} by multiplying \frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}} by the reciprocal of \frac{\left(a^{2}\right)^{-2}}{\left(5b^{3}\right)^{-2}}.

$$\frac{5^{-4}\times \left(5b^{3}\right)^{-2}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Calculate 5 to the power of -4 and get \frac{1}{625}.

$$\frac{\frac{1}{625}\times \left(5b^{3}\right)^{-2}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Expand \left(5b^{3}\right)^{-2}.

$$\frac{\frac{1}{625}\times 5^{-2}\left(b^{3}\right)^{-2}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{\frac{1}{625}\times 5^{-2}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\frac{1}{625}\times \frac{1}{25}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Multiply \frac{1}{625} and \frac{1}{25} to get \frac{1}{15625}.

$$\frac{\frac{1}{15625}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -2 to get -4.

$$\frac{\frac{1}{15625}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}a^{-4}}$$

Expand \left(4a^{\frac{5}{3}}b^{2}\right)^{-4}.

$$\frac{\frac{1}{15625}b^{-6}}{4^{-4}\left(a^{\frac{5}{3}}\right)^{-4}\left(b^{2}\right)^{-4}a^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{3} and -4 to get -\frac{20}{3}.

$$\frac{\frac{1}{15625}b^{-6}}{4^{-4}a^{-\frac{20}{3}}\left(b^{2}\right)^{-4}a^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -4 to get -8.

$$\frac{\frac{1}{15625}b^{-6}}{4^{-4}a^{-\frac{20}{3}}b^{-8}a^{-4}}$$

Calculate 4 to the power of -4 and get \frac{1}{256}.

$$\frac{\frac{1}{15625}b^{-6}}{\frac{1}{256}a^{-\frac{20}{3}}b^{-8}a^{-4}}$$

To multiply powers of the same base, add their exponents. Add -\frac{20}{3} and -4 to get -\frac{32}{3}.

$$\frac{\frac{1}{15625}b^{-6}}{\frac{1}{256}a^{-\frac{32}{3}}b^{-8}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{1}{15625}b^{2}}{\frac{1}{256}a^{-\frac{32}{3}}}$$

Expand

$$\frac{256b^{2}a^{\frac{32}{3}}}{15625}$$

Solution Steps

Cancel out \sqrt[3]{a}b^{3} in both numerator and denominator.

$$\frac{\left(\frac{5}{4a^{\frac{5}{3}}b^{2}}\right)^{-4}}{\left(\frac{a^{2}}{5b^{3}}\right)^{-2}}$$

To raise \frac{5}{4a^{\frac{5}{3}}b^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}}}{\left(\frac{a^{2}}{5b^{3}}\right)^{-2}}$$

To raise \frac{a^{2}}{5b^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}}}{\frac{\left(a^{2}\right)^{-2}}{\left(5b^{3}\right)^{-2}}}$$

Divide \frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}} by \frac{\left(a^{2}\right)^{-2}}{\left(5b^{3}\right)^{-2}} by multiplying \frac{5^{-4}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}} by the reciprocal of \frac{\left(a^{2}\right)^{-2}}{\left(5b^{3}\right)^{-2}}.

$$\frac{5^{-4}\times \left(5b^{3}\right)^{-2}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Calculate 5 to the power of -4 and get \frac{1}{625}.

$$\frac{\frac{1}{625}\times \left(5b^{3}\right)^{-2}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Expand \left(5b^{3}\right)^{-2}.

$$\frac{\frac{1}{625}\times 5^{-2}\left(b^{3}\right)^{-2}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{\frac{1}{625}\times 5^{-2}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\frac{1}{625}\times \frac{1}{25}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

Multiply \frac{1}{625} and \frac{1}{25} to get \frac{1}{15625}.

$$\frac{\frac{1}{15625}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}\left(a^{2}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -2 to get -4.

$$\frac{\frac{1}{15625}b^{-6}}{\left(4a^{\frac{5}{3}}b^{2}\right)^{-4}a^{-4}}$$

Expand \left(4a^{\frac{5}{3}}b^{2}\right)^{-4}.

$$\frac{\frac{1}{15625}b^{-6}}{4^{-4}\left(a^{\frac{5}{3}}\right)^{-4}\left(b^{2}\right)^{-4}a^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{3} and -4 to get -\frac{20}{3}.

$$\frac{\frac{1}{15625}b^{-6}}{4^{-4}a^{-\frac{20}{3}}\left(b^{2}\right)^{-4}a^{-4}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -4 to get -8.

$$\frac{\frac{1}{15625}b^{-6}}{4^{-4}a^{-\frac{20}{3}}b^{-8}a^{-4}}$$

Calculate 4 to the power of -4 and get \frac{1}{256}.

$$\frac{\frac{1}{15625}b^{-6}}{\frac{1}{256}a^{-\frac{20}{3}}b^{-8}a^{-4}}$$

To multiply powers of the same base, add their exponents. Add -\frac{20}{3} and -4 to get -\frac{32}{3}.

$$\frac{\frac{1}{15625}b^{-6}}{\frac{1}{256}a^{-\frac{32}{3}}b^{-8}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\frac{1}{15625}b^{2}}{\frac{1}{256}a^{-\frac{32}{3}}}$$