Solve (a/2b^2)^3div(3a/b)^-2

$$( \frac { a } { 2 b ^ { 2 } } ) ^ { 3 } \div ( \frac { 3 a } { b } ) ^ { - 2 }$$

Evaluate

$$\frac{9a^{5}}{8b^{8}}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Calculate 2 to the power of 3 and get 8.

$$\frac{a^{3}b^{-2}}{8b^{6}\times \left(3a\right)^{-2}}$$

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

$$\frac{a^{3}b^{-2}}{8b^{6}\times 3^{-2}a^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{a^{3}b^{-2}}{8b^{6}\times \frac{1}{9}a^{-2}}$$

Multiply 8 and \frac{1}{9} to get \frac{8}{9}.

$$\frac{a^{3}b^{-2}}{\frac{8}{9}b^{6}a^{-2}}$$

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

$$\frac{b^{-2}a^{5}}{\frac{8}{9}b^{6}}$$

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

$$\frac{a^{5}}{\frac{8}{9}b^{8}}$$

Expand

$$\frac{9a^{5}}{8b^{8}}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Calculate 2 to the power of 3 and get 8.

$$\frac{a^{3}b^{-2}}{8b^{6}\times \left(3a\right)^{-2}}$$

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

$$\frac{a^{3}b^{-2}}{8b^{6}\times 3^{-2}a^{-2}}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{a^{3}b^{-2}}{8b^{6}\times \frac{1}{9}a^{-2}}$$

Multiply 8 and \frac{1}{9} to get \frac{8}{9}.

$$\frac{a^{3}b^{-2}}{\frac{8}{9}b^{6}a^{-2}}$$

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

$$\frac{b^{-2}a^{5}}{\frac{8}{9}b^{6}}$$

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

$$\frac{a^{5}}{\frac{8}{9}b^{8}}$$