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

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

Evaluate

$$\frac{b^{2}}{12a}$$

Solution Steps

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

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

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

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

Multiply \frac{\left(2a^{2}\right)^{-2}}{\left(3b\right)^{-2}} times \frac{3^{-3}}{a^{-3}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

Calculate 2 to the power of -2 and get \frac{1}{4}.

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

Calculate 3 to the power of -3 and get \frac{1}{27}.

$$\frac{\frac{1}{4}a^{-4}\times \frac{1}{27}}{\left(3b\right)^{-2}a^{-3}}$$

Multiply \frac{1}{4} and \frac{1}{27} to get \frac{1}{108}.

$$\frac{\frac{1}{108}a^{-4}}{\left(3b\right)^{-2}a^{-3}}$$

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

$$\frac{\frac{1}{108}a^{-4}}{3^{-2}b^{-2}a^{-3}}$$

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

$$\frac{\frac{1}{108}a^{-4}}{\frac{1}{9}b^{-2}a^{-3}}$$

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

$$\frac{\frac{1}{108}}{\frac{1}{9}b^{-2}a^{1}}$$

Express \frac{\frac{1}{108}}{\frac{1}{9}b^{-2}a^{1}} as a single fraction.

$$\frac{1}{108\times \frac{1}{9}b^{-2}a^{1}}$$

Multiply 108 and \frac{1}{9} to get 12.

$$\frac{1}{12b^{-2}a^{1}}$$

Calculate a to the power of 1 and get a.

$$\frac{1}{12b^{-2}a}$$

Expand

$$\frac{b^{2}}{12a}$$

Solution Steps

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

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

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

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

Multiply \frac{\left(2a^{2}\right)^{-2}}{\left(3b\right)^{-2}} times \frac{3^{-3}}{a^{-3}} by multiplying numerator times numerator and denominator times denominator.

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

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

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

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

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

Calculate 2 to the power of -2 and get \frac{1}{4}.

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

Calculate 3 to the power of -3 and get \frac{1}{27}.

$$\frac{\frac{1}{4}a^{-4}\times \frac{1}{27}}{\left(3b\right)^{-2}a^{-3}}$$

Multiply \frac{1}{4} and \frac{1}{27} to get \frac{1}{108}.

$$\frac{\frac{1}{108}a^{-4}}{\left(3b\right)^{-2}a^{-3}}$$

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

$$\frac{\frac{1}{108}a^{-4}}{3^{-2}b^{-2}a^{-3}}$$

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

$$\frac{\frac{1}{108}a^{-4}}{\frac{1}{9}b^{-2}a^{-3}}$$

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

$$\frac{\frac{1}{108}}{\frac{1}{9}b^{-2}a^{1}}$$

Express \frac{\frac{1}{108}}{\frac{1}{9}b^{-2}a^{1}} as a single fraction.

$$\frac{1}{108\times \frac{1}{9}b^{-2}a^{1}}$$

Multiply 108 and \frac{1}{9} to get 12.

$$\frac{1}{12b^{-2}a^{1}}$$

Calculate a to the power of 1 and get a.

$$\frac{1}{12b^{-2}a}$$