Solve [2(-b)^{-3}]^{-2}-[4(-b/2)^{2}]^{3}+(-2)^11*[-(4b^-1)^-2]^3

$$[ 2 ( - b ) ^ { - 3 } ] ^ { - 2 } - [ 4 ( - \frac { b } { 2 } ) ^ { 2 } ] ^ { 3 } + ( - 2 ) ^ { 11 } \cdot [ - ( 4 b ^ { - 1 } ) ^ { - 2 } ] ^ { 3 }$$

Evaluate

$$-\frac{b^{6}}{4}$$

Solution Steps

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

$$2^{-2}\left(\left(-b\right)^{-3}\right)^{-2}-\left(4\left(-\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$2^{-2}\left(-b\right)^{6}-\left(4\left(-\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-\left(4\left(-\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate -\frac{b}{2} to the power of 2 and get \left(\frac{b}{2}\right)^{2}.

$$\frac{1}{4}\left(-b\right)^{6}-\left(4\times \left(\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Expand \left(4\times \left(\frac{b}{2}\right)^{2}\right)^{3}.

$$\frac{1}{4}\left(-b\right)^{6}-4^{3}\times \left(\left(\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-4^{3}\times \left(\frac{b}{2}\right)^{6}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{1}{4}\left(-b\right)^{6}-64\times \left(\frac{b}{2}\right)^{6}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-64\times \frac{b^{6}}{2^{6}}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Express 64\times \frac{b^{6}}{2^{6}} as a single fraction.

$$\frac{1}{4}\left(-b\right)^{6}-\frac{64b^{6}}{2^{6}}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{1}{4}\left(-b\right)^{6}-\frac{64b^{6}}{64}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Cancel out 64 and 64.

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate -2 to the power of 11 and get -2048.

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Expand \left(4b^{-1}\right)^{-2}.

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-4^{-2}\left(b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-4^{-2}b^{2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Expand \left(-b\right)^{6}.

$$\frac{1}{4}\left(-1\right)^{6}b^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Calculate -1 to the power of 6 and get 1.

$$\frac{1}{4}\times 1b^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

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

$$\frac{1}{4}b^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Combine \frac{1}{4}b^{6} and -b^{6} to get -\frac{3}{4}b^{6}.

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Expand \left(-\frac{1}{16}b^{2}\right)^{3}.

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{16}\right)^{3}\left(b^{2}\right)^{3}$$

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

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{16}\right)^{3}b^{6}$$

Calculate -\frac{1}{16} to the power of 3 and get -\frac{1}{4096}.

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{4096}\right)b^{6}$$

Multiply -2048 and -\frac{1}{4096} to get \frac{1}{2}.

$$-\frac{3}{4}b^{6}+\frac{1}{2}b^{6}$$

Combine -\frac{3}{4}b^{6} and \frac{1}{2}b^{6} to get -\frac{1}{4}b^{6}.

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

Expand

$$-\frac{b^{6}}{4}$$

Solution Steps

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

$$2^{-2}\left(\left(-b\right)^{-3}\right)^{-2}-\left(4\left(-\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$2^{-2}\left(-b\right)^{6}-\left(4\left(-\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-\left(4\left(-\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate -\frac{b}{2} to the power of 2 and get \left(\frac{b}{2}\right)^{2}.

$$\frac{1}{4}\left(-b\right)^{6}-\left(4\times \left(\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Expand \left(4\times \left(\frac{b}{2}\right)^{2}\right)^{3}.

$$\frac{1}{4}\left(-b\right)^{6}-4^{3}\times \left(\left(\frac{b}{2}\right)^{2}\right)^{3}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-4^{3}\times \left(\frac{b}{2}\right)^{6}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate 4 to the power of 3 and get 64.

$$\frac{1}{4}\left(-b\right)^{6}-64\times \left(\frac{b}{2}\right)^{6}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-64\times \frac{b^{6}}{2^{6}}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Express 64\times \frac{b^{6}}{2^{6}} as a single fraction.

$$\frac{1}{4}\left(-b\right)^{6}-\frac{64b^{6}}{2^{6}}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate 2 to the power of 6 and get 64.

$$\frac{1}{4}\left(-b\right)^{6}-\frac{64b^{6}}{64}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Cancel out 64 and 64.

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}+\left(-2\right)^{11}\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Calculate -2 to the power of 11 and get -2048.

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-\left(4b^{-1}\right)^{-2}\right)^{3}$$

Expand \left(4b^{-1}\right)^{-2}.

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-4^{-2}\left(b^{-1}\right)^{-2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-4^{-2}b^{2}\right)^{3}$$

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

$$\frac{1}{4}\left(-b\right)^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Expand \left(-b\right)^{6}.

$$\frac{1}{4}\left(-1\right)^{6}b^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Calculate -1 to the power of 6 and get 1.

$$\frac{1}{4}\times 1b^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

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

$$\frac{1}{4}b^{6}-b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Combine \frac{1}{4}b^{6} and -b^{6} to get -\frac{3}{4}b^{6}.

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{16}b^{2}\right)^{3}$$

Expand \left(-\frac{1}{16}b^{2}\right)^{3}.

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{16}\right)^{3}\left(b^{2}\right)^{3}$$

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

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{16}\right)^{3}b^{6}$$

Calculate -\frac{1}{16} to the power of 3 and get -\frac{1}{4096}.

$$-\frac{3}{4}b^{6}-2048\left(-\frac{1}{4096}\right)b^{6}$$

Multiply -2048 and -\frac{1}{4096} to get \frac{1}{2}.

$$-\frac{3}{4}b^{6}+\frac{1}{2}b^{6}$$

Combine -\frac{3}{4}b^{6} and \frac{1}{2}b^{6} to get -\frac{1}{4}b^{6}.

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