Solve [(9^{5}:3^{8})x-3^{0}][(16^{6}:4^{11})x+2]=[6x+(5^7)^3:25^10:5]^2

$$[ ( 9 ^ { 5 } : 3 ^ { 8 } ) x - 3 ^ { 0 } ] [ ( 16 ^ { 6 } : 4 ^ { 11 } ) x + 2 ] = [ 6 x + ( 5 ^ { 7 } ) ^ { 3 } : 25 ^ { 10 } : 5 ] ^ { 2 }$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps for Solving Linear Equation

To raise a power to another power, multiply the exponents. Multiply 7 and 3 to get 21.

$$\left(\frac{9^{5}}{3^{8}}x-3^{0}\right)\left(\frac{16^{6}}{4^{11}}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Calculate 9 to the power of 5 and get 59049.

$$\left(\frac{59049}{3^{8}}x-3^{0}\right)\left(\frac{16^{6}}{4^{11}}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Calculate 3 to the power of 8 and get 6561.

$$\left(\frac{59049}{6561}x-3^{0}\right)\left(\frac{16^{6}}{4^{11}}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Divide 59049 by 6561 to get 9.

$$\left(9x-3^{0}\right)\left(\frac{16^{6}}{4^{11}}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Calculate 3 to the power of 0 and get 1.

$$\left(9x-1\right)\left(\frac{16^{6}}{4^{11}}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Calculate 16 to the power of 6 and get 16777216.

$$\left(9x-1\right)\left(\frac{16777216}{4^{11}}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Calculate 4 to the power of 11 and get 4194304.

$$\left(9x-1\right)\left(\frac{16777216}{4194304}x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Divide 16777216 by 4194304 to get 4.

$$\left(9x-1\right)\left(4x+2\right)=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Use the distributive property to multiply 9x-1 by 4x+2 and combine like terms.

$$36x^{2}+14x-2=\left(6x+\frac{\frac{5^{21}}{25^{10}}}{5}\right)^{2}$$

Calculate 5 to the power of 21 and get 476837158203125.

$$36x^{2}+14x-2=\left(6x+\frac{\frac{476837158203125}{25^{10}}}{5}\right)^{2}$$

Calculate 25 to the power of 10 and get 95367431640625.

$$36x^{2}+14x-2=\left(6x+\frac{\frac{476837158203125}{95367431640625}}{5}\right)^{2}$$

Divide 476837158203125 by 95367431640625 to get 5.

$$36x^{2}+14x-2=\left(6x+\frac{5}{5}\right)^{2}$$

Divide 5 by 5 to get 1.

$$36x^{2}+14x-2=\left(6x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6x+1\right)^{2}.

$$36x^{2}+14x-2=36x^{2}+12x+1$$

Subtract 36x^{2} from both sides.

$$36x^{2}+14x-2-36x^{2}=12x+1$$

Combine 36x^{2} and -36x^{2} to get 0.

$$14x-2=12x+1$$

Subtract 12x from both sides.

$$14x-2-12x=1$$

Combine 14x and -12x to get 2x.

$$2x-2=1$$

Add 2 to both sides.

$$2x=1+2$$

Add 1 and 2 to get 3.

$$2x=3$$

Divide both sides by 2.

$$x=\frac{3}{2}$$