Solve (4ab)^-1div(a^2b^-3/6)^-2

$$( 4 a b ) ^ { - 1 } \div ( \frac { a ^ { 2 } b ^ { - 3 } } { 6 } ) ^ { - 2 }$$

Evaluate

$$\frac{a^{3}}{144b^{7}}$$

Solution Steps

Expand \left(4ab\right)^{-1}.

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

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

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

To raise \frac{a^{2}b^{-3}}{6} to a power, raise both numerator and denominator to the power and then divide.

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

Divide \frac{1}{4}a^{-1}b^{-1} by \frac{\left(a^{2}b^{-3}\right)^{-2}}{6^{-2}} by multiplying \frac{1}{4}a^{-1}b^{-1} by the reciprocal of \frac{\left(a^{2}b^{-3}\right)^{-2}}{6^{-2}}.

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

Calculate 6 to the power of -2 and get \frac{1}{36}.

$$\frac{\frac{1}{4}a^{-1}b^{-1}\times \frac{1}{36}}{\left(a^{2}b^{-3}\right)^{-2}}$$

Multiply \frac{1}{4} and \frac{1}{36} to get \frac{1}{144}.

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{\left(a^{2}b^{-3}\right)^{-2}}$$

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

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{\left(a^{2}\right)^{-2}\left(b^{-3}\right)^{-2}}$$

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

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{a^{-4}\left(b^{-3}\right)^{-2}}$$

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

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{a^{-4}b^{6}}$$

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

$$\frac{\frac{1}{144}\times \frac{1}{b}a^{3}}{b^{6}}$$

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

$$\frac{\frac{1}{144}a^{3}}{b^{7}}$$

Expand

$$\frac{a^{3}}{144b^{7}}$$

Solution Steps

Expand \left(4ab\right)^{-1}.

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

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

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

To raise \frac{a^{2}b^{-3}}{6} to a power, raise both numerator and denominator to the power and then divide.

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

Divide \frac{1}{4}a^{-1}b^{-1} by \frac{\left(a^{2}b^{-3}\right)^{-2}}{6^{-2}} by multiplying \frac{1}{4}a^{-1}b^{-1} by the reciprocal of \frac{\left(a^{2}b^{-3}\right)^{-2}}{6^{-2}}.

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

Calculate 6 to the power of -2 and get \frac{1}{36}.

$$\frac{\frac{1}{4}a^{-1}b^{-1}\times \frac{1}{36}}{\left(a^{2}b^{-3}\right)^{-2}}$$

Multiply \frac{1}{4} and \frac{1}{36} to get \frac{1}{144}.

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{\left(a^{2}b^{-3}\right)^{-2}}$$

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

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{\left(a^{2}\right)^{-2}\left(b^{-3}\right)^{-2}}$$

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

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{a^{-4}\left(b^{-3}\right)^{-2}}$$

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

$$\frac{\frac{1}{144}a^{-1}b^{-1}}{a^{-4}b^{6}}$$

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

$$\frac{\frac{1}{144}\times \frac{1}{b}a^{3}}{b^{6}}$$

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

$$\frac{\frac{1}{144}a^{3}}{b^{7}}$$