Solve ((2b^-2)^4/3b^3)^3

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

Evaluate

$$\frac{4096}{27b^{33}}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 4 and get 16.

$$\left(\frac{16b^{-8}}{3b^{3}}\right)^{3}$$

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

$$\left(\frac{16}{3b^{11}}\right)^{3}$$

To raise \frac{16}{3b^{11}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{16^{3}}{\left(3b^{11}\right)^{3}}$$

Calculate 16 to the power of 3 and get 4096.

$$\frac{4096}{\left(3b^{11}\right)^{3}}$$

Expand \left(3b^{11}\right)^{3}.

$$\frac{4096}{3^{3}\left(b^{11}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 11 and 3 to get 33.

$$\frac{4096}{3^{3}b^{33}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4096}{27b^{33}}$$

Expand

$$\frac{4096}{27b^{33}}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 4 and get 16.

$$\left(\frac{16b^{-8}}{3b^{3}}\right)^{3}$$

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

$$\left(\frac{16}{3b^{11}}\right)^{3}$$

To raise \frac{16}{3b^{11}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{16^{3}}{\left(3b^{11}\right)^{3}}$$

Calculate 16 to the power of 3 and get 4096.

$$\frac{4096}{\left(3b^{11}\right)^{3}}$$

Expand \left(3b^{11}\right)^{3}.

$$\frac{4096}{3^{3}\left(b^{11}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 11 and 3 to get 33.

$$\frac{4096}{3^{3}b^{33}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4096}{27b^{33}}$$