`@` `\text {Ans}`

`\downarrow`

`3^4*5^2 - 128*2^3 + 1^17`

`= 9^2*5^2 - 2^7*2^3 + 1`

`= (9*5)^2 - 2^10+1`

`= 45^5-2^10 + 1`

`= 2025 - 1024 + 1`

`= 2025 - 1023`

`= 1002`

\(24-4^2\div4\cdot2+3\)

`= 24 - 4*2 + 3`

`= 24 - 8 + 3`

`= 24 - 5`

`= 19`

`@` `\text {Kaizuu lv u}.`

\(D=4\times5^{100}\times\left[\left(\frac{1}{5}\right)\left(\frac{1}{5}\right)^2\left(\frac{1}{5}\right)^3...\left(\frac{1}{5}\right)^{100}\right]+1\)

\(=4\times5^{100}\times\frac{1}{5\times5^2\times...\times5^{100}}+1\)

\(=\frac{4}{5\times5^2\times...\times5^{99}}+1\)

\(=\frac{4}{5^{99\left(1+99\right):2}}+1=\frac{4}{5^{4950}}+1\)

a) \(\left(3x-2\right)\left(3x-1\right)=\left(3x+1\right)^2\)

<=> \(9x^2-9x+2=9x^2+6x+1\)

<=>  \(15x=1\) <=> \(x=\frac{1}{15}\)

b) \(\left(4x-1\right)\left(x+1\right)=\left(2x-3\right)^2\)

<=> \(4x^2+3x-1=4x^2-12x+9\)

<=> \(15x^2=10\) <=> \(x=\frac{2}{3}\)

c) \(\left(5x+1\right)^2=\left(7x-3\right)\left(7x+2\right)\) <=> \(25x^2+10x+1=49x^2-7x-6\)

<=> \(24x^2-17x-7=0\) <=> \(24x^2-24x+7x-7=0\)

<=> \(\left(24x+7\right)\left(x-1\right)=0\) <=> \(\orbr{\begin{cases}x=-\frac{7}{24}\\x=1\end{cases}}\)

d) (4 - 3x)(4 + 3x) = (9x - 3)(1 - x)

<=> 16 - 9x² = 12x - 9x2 - 3

<=> 12x = 19

<=> x = 19/12

e) x(x + 1)(x + 2)(x + 3) = 24

<=> (x² + 3x)(x² + 3x + 2) = 24

<=> (x² + 3x)²  + 2(x² + 3x) - 24 = 0

<=> (x² + 3x)² + 6(x² + 3x) - 4(x² + 3x) - 24 = 0

<=> (x² + 3x + 6)(x² + 3x - 4) = 0

<=> \(\orbr{\begin{cases}x^2+3x+6=0\\x^2+3x-4=0\end{cases}}\)

<=> \(\orbr{\begin{cases}\left(x+\frac{3}{2}\right)^2+\frac{15}{4}=0\left(vn\right)\\\left(x+4\right)\left(x-1\right)=0\end{cases}}\)

<=> \(\orbr{\begin{cases}x=-4\\x=1\end{cases}}\)

g) (7x - 2)² = (7x - 3)(7x + 2)

<=> 49x² - 28x + 4 = 49x² - 7x - 6

<=> 21x = 10 <=> x = 10/21

\(\left(7x+3\right)^2-\left(7x-1\right)\left(7x-3\right)=-12\)

\(\Rightarrow49x^2+42x+9-\left(49x^2-21x-7x+3\right)=-12\)

\(\Rightarrow70x+18=0\) \(\Rightarrow x=-\dfrac{18}{70}=-\dfrac{9}{35}\)