Solve {[(2^{5})^{3}:4^{7}]x-1}^{2}=(2x+2^{0})^2+(2^7)^5:4^15

$$\{ [ ( 2 ^ { 5 } ) ^ { 3 } : 4 ^ { 7 } ] x - 1 \} ^ { 2 } = ( 2 x + 2 ^ { 0 } ) ^ { 2 } + ( 2 ^ { 7 } ) ^ { 5 } : 4 ^ { 15 }$$

Solve for x

$$x=-4$$

Steps for Solving Linear Equation

To raise a power to another power, multiply the exponents. Multiply 5 and 3 to get 15.

$$\left(\frac{2^{15}}{4^{7}}x-1\right)^{2}=\left(2x+2^{0}\right)^{2}+\frac{\left(2^{7}\right)^{5}}{4^{15}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and 5 to get 35.

$$\left(\frac{2^{15}}{4^{7}}x-1\right)^{2}=\left(2x+2^{0}\right)^{2}+\frac{2^{35}}{4^{15}}$$

Calculate 2 to the power of 15 and get 32768.

$$\left(\frac{32768}{4^{7}}x-1\right)^{2}=\left(2x+2^{0}\right)^{2}+\frac{2^{35}}{4^{15}}$$

Calculate 4 to the power of 7 and get 16384.

$$\left(\frac{32768}{16384}x-1\right)^{2}=\left(2x+2^{0}\right)^{2}+\frac{2^{35}}{4^{15}}$$

Divide 32768 by 16384 to get 2.

$$\left(2x-1\right)^{2}=\left(2x+2^{0}\right)^{2}+\frac{2^{35}}{4^{15}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-1\right)^{2}.

$$4x^{2}-4x+1=\left(2x+2^{0}\right)^{2}+\frac{2^{35}}{4^{15}}$$

Calculate 2 to the power of 0 and get 1.

$$4x^{2}-4x+1=\left(2x+1\right)^{2}+\frac{2^{35}}{4^{15}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$4x^{2}-4x+1=4x^{2}+4x+1+\frac{2^{35}}{4^{15}}$$

Calculate 2 to the power of 35 and get 34359738368.

$$4x^{2}-4x+1=4x^{2}+4x+1+\frac{34359738368}{4^{15}}$$

Calculate 4 to the power of 15 and get 1073741824.

$$4x^{2}-4x+1=4x^{2}+4x+1+\frac{34359738368}{1073741824}$$

Divide 34359738368 by 1073741824 to get 32.

$$4x^{2}-4x+1=4x^{2}+4x+1+32$$

Add 1 and 32 to get 33.

$$4x^{2}-4x+1=4x^{2}+4x+33$$

Subtract 4x^{2} from both sides.

$$4x^{2}-4x+1-4x^{2}=4x+33$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-4x+1=4x+33$$

Subtract 4x from both sides.

$$-4x+1-4x=33$$

Combine -4x and -4x to get -8x.

$$-8x+1=33$$

Subtract 1 from both sides.

$$-8x=33-1$$

Subtract 1 from 33 to get 32.

$$-8x=32$$

Divide both sides by -8.

$$x=\frac{32}{-8}$$

Divide 32 by -8 to get -4.

$$x=-4$$