Solve [7^{0}+sqrt{frac{5^{2}-2^{5}}{-(2^{3}-3^{-2}*3^{2})}-1}]^{5}-[(1/4)^3:(1/4)^2]^-1*sqrt[3]{-frac{1/64}}{(-2)^-3}

$$[ 7 ^ { 0 } + \sqrt { \frac { 5 ^ { 2 } - 2 ^ { 5 } } { - ( 2 ^ { 3 } - 3 ^ { - 2 } \cdot 3 ^ { 2 } ) } - 1 } ] ^ { 5 } - [ ( \frac { 1 } { 4 } ) ^ { 3 } : ( \frac { 1 } { 4 } ) ^ { 2 } ] ^ { - 1 } \cdot \frac { \sqrt[ 3 ] { - \frac { 1 } { 64 } } } { ( - 2 ) ^ { - 3 } }$$

Evaluate

$$-7$$

Solution Steps

Multiply 3^{-2} and 3^{2} to get 1.

$$\left(7^{0}+\sqrt{\frac{5^{2}-2^{5}}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\frac{\left(\frac{1}{4}\right)^{3}}{\left(\frac{1}{4}\right)^{2}}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 2 from 3 to get 1.

$$\left(7^{0}+\sqrt{\frac{5^{2}-2^{5}}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\left(\frac{1}{4}\right)^{1}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

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

$$\left(7^{0}+\sqrt{\frac{5^{2}-2^{5}}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate 7 to the power of 0 and get 1.

$$\left(1+\sqrt{\frac{5^{2}-2^{5}}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate 5 to the power of 2 and get 25.

$$\left(1+\sqrt{\frac{25-2^{5}}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate 2 to the power of 5 and get 32.

$$\left(1+\sqrt{\frac{25-32}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Subtract 32 from 25 to get -7.

$$\left(1+\sqrt{\frac{-7}{-\left(2^{3}-1\right)}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate 2 to the power of 3 and get 8.

$$\left(1+\sqrt{\frac{-7}{-\left(8-1\right)}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Subtract 1 from 8 to get 7.

$$\left(1+\sqrt{\frac{-7}{-7}-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Divide -7 by -7 to get 1.

$$\left(1+\sqrt{1-1}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Subtract 1 from 1 to get 0.

$$\left(1+\sqrt{0}\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate the square root of 0 and get 0.

$$\left(1+0\right)^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Add 1 and 0 to get 1.

$$1^{5}-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate 1 to the power of 5 and get 1.

$$1-\left(\frac{1}{4}\right)^{-1}\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

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

$$1-4\times \frac{\sqrt[3]{-\frac{1}{64}}}{\left(-2\right)^{-3}}$$

Calculate \sqrt[3]{-\frac{1}{64}} and get -\frac{1}{4}.

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

Calculate -2 to the power of -3 and get -\frac{1}{8}.

$$1-4\times \frac{-\frac{1}{4}}{-\frac{1}{8}}$$

Divide -\frac{1}{4} by -\frac{1}{8} by multiplying -\frac{1}{4} by the reciprocal of -\frac{1}{8}.

$$1-4\left(-\frac{1}{4}\right)\left(-8\right)$$

Multiply -\frac{1}{4} and -8 to get 2.

$$1-4\times 2$$

Multiply 4 and 2 to get 8.

$$1-8$$

Subtract 8 from 1 to get -7.

$$-7$$

Factor

$$-7$$