Solve frac{(3^{-1}a^{-1}b^{-2})^{-2}*(4a^{-3}b^{4})^-2}{(3a^-3b^-4)^2}

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

Evaluate

$$\frac{b^{4}a^{14}}{16}$$

Solution Steps

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

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

Expand \left(\frac{1}{3}a^{-1}b^{-2}\right)^{-2}.

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

To raise a power to another power, multiply the exponents. Multiply -1 and -2 to get 2.

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

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

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

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

$$\frac{9a^{2}b^{4}\times \left(4a^{-3}b^{4}\right)^{-2}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{9a^{2}b^{4}\times 4^{-2}\left(a^{-3}\right)^{-2}\left(b^{4}\right)^{-2}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{9a^{2}b^{4}\times 4^{-2}a^{6}\left(b^{4}\right)^{-2}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{9a^{2}b^{4}\times 4^{-2}a^{6}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{9a^{2}b^{4}\times \frac{1}{16}a^{6}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

Multiply 9 and \frac{1}{16} to get \frac{9}{16}.

$$\frac{\frac{9}{16}a^{2}b^{4}a^{6}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$\frac{\frac{9}{16}a^{8}b^{4}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and -8 to get -4.

$$\frac{\frac{9}{16}a^{8}b^{-4}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{\frac{9}{16}a^{8}b^{-4}}{3^{2}\left(a^{-3}\right)^{2}\left(b^{-4}\right)^{2}}$$

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

$$\frac{\frac{9}{16}a^{8}b^{-4}}{3^{2}a^{-6}\left(b^{-4}\right)^{2}}$$

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

$$\frac{\frac{9}{16}a^{8}b^{-4}}{3^{2}a^{-6}b^{-8}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{9}{16}a^{8}b^{-4}}{9a^{-6}b^{-8}}$$

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

$$\frac{\frac{9}{16}b^{4}a^{14}}{9}$$

Divide \frac{9}{16}b^{4}a^{14} by 9 to get \frac{1}{16}b^{4}a^{14}.

$$\frac{1}{16}b^{4}a^{14}$$

Expand

$$\frac{b^{4}a^{14}}{16}$$

Solution Steps

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

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

Expand \left(\frac{1}{3}a^{-1}b^{-2}\right)^{-2}.

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

To raise a power to another power, multiply the exponents. Multiply -1 and -2 to get 2.

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

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

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

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

$$\frac{9a^{2}b^{4}\times \left(4a^{-3}b^{4}\right)^{-2}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{9a^{2}b^{4}\times 4^{-2}\left(a^{-3}\right)^{-2}\left(b^{4}\right)^{-2}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{9a^{2}b^{4}\times 4^{-2}a^{6}\left(b^{4}\right)^{-2}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{9a^{2}b^{4}\times 4^{-2}a^{6}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

Calculate 4 to the power of -2 and get \frac{1}{16}.

$$\frac{9a^{2}b^{4}\times \frac{1}{16}a^{6}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

Multiply 9 and \frac{1}{16} to get \frac{9}{16}.

$$\frac{\frac{9}{16}a^{2}b^{4}a^{6}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 6 to get 8.

$$\frac{\frac{9}{16}a^{8}b^{4}b^{-8}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and -8 to get -4.

$$\frac{\frac{9}{16}a^{8}b^{-4}}{\left(3a^{-3}b^{-4}\right)^{2}}$$

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

$$\frac{\frac{9}{16}a^{8}b^{-4}}{3^{2}\left(a^{-3}\right)^{2}\left(b^{-4}\right)^{2}}$$

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

$$\frac{\frac{9}{16}a^{8}b^{-4}}{3^{2}a^{-6}\left(b^{-4}\right)^{2}}$$

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

$$\frac{\frac{9}{16}a^{8}b^{-4}}{3^{2}a^{-6}b^{-8}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\frac{9}{16}a^{8}b^{-4}}{9a^{-6}b^{-8}}$$

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

$$\frac{\frac{9}{16}b^{4}a^{14}}{9}$$

Divide \frac{9}{16}b^{4}a^{14} by 9 to get \frac{1}{16}b^{4}a^{14}.

$$\frac{1}{16}b^{4}a^{14}$$