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

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

Evaluate

$$\frac{256}{a^{4}b^{12}}$$

Solution Steps

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

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

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

$$\left(\frac{\frac{1}{16}a^{1}b^{3}}{\frac{1}{4}}\right)^{-4}$$

Calculate a to the power of 1 and get a.

$$\left(\frac{\frac{1}{16}ab^{3}}{\frac{1}{4}}\right)^{-4}$$

Divide \frac{1}{16}ab^{3} by \frac{1}{4} by multiplying \frac{1}{16}ab^{3} by the reciprocal of \frac{1}{4}.

$$\left(\frac{1}{16}ab^{3}\times 4\right)^{-4}$$

Multiply \frac{1}{16} and 4 to get \frac{1}{4}.

$$\left(\frac{1}{4}ab^{3}\right)^{-4}$$

Expand \left(\frac{1}{4}ab^{3}\right)^{-4}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\left(\frac{1}{4}\right)^{-4}a^{-4}b^{-12}$$

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

$$256a^{-4}b^{-12}$$

Expand

$$\frac{256}{a^{4}b^{12}}$$

Solution Steps

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

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

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

$$\left(\frac{\frac{1}{16}a^{1}b^{3}}{\frac{1}{4}}\right)^{-4}$$

Calculate a to the power of 1 and get a.

$$\left(\frac{\frac{1}{16}ab^{3}}{\frac{1}{4}}\right)^{-4}$$

Divide \frac{1}{16}ab^{3} by \frac{1}{4} by multiplying \frac{1}{16}ab^{3} by the reciprocal of \frac{1}{4}.

$$\left(\frac{1}{16}ab^{3}\times 4\right)^{-4}$$

Multiply \frac{1}{16} and 4 to get \frac{1}{4}.

$$\left(\frac{1}{4}ab^{3}\right)^{-4}$$

Expand \left(\frac{1}{4}ab^{3}\right)^{-4}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\left(\frac{1}{4}\right)^{-4}a^{-4}b^{-12}$$

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

$$256a^{-4}b^{-12}$$