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Solve the exponential equation for 
x.

{:[2^(5x+8)=((1)/(32))^(3)],[x=◻]:}

Solve the exponential equation for x x .\newline25x+8=(132)3x= \begin{array}{l} 2^{5 x+8}=\left(\frac{1}{32}\right)^{3} \\ x=\square \end{array}

Full solution

Q. Solve the exponential equation for x x .\newline25x+8=(132)3x= \begin{array}{l} 2^{5 x+8}=\left(\frac{1}{32}\right)^{3} \\ x=\square \end{array}
  1. Write equation: Write down the given exponential equation.\newline25x+8=(132)32^{5x+8} = \left(\frac{1}{32}\right)^3
  2. Recognize power of 22: Recognize that 132\frac{1}{32} is a power of 22, since 32=2532 = 2^5. Therefore, (132)3=(25)3(\frac{1}{32})^3 = (2^{-5})^3.
  3. Simplify right side: Simplify the right side of the equation using the power of a power rule (ab)c=a(bc)(a^b)^c = a^{(b*c)}.\newline(25)3=2(53)=215(2^{-5})^3 = 2^{(-5*3)} = 2^{-15}
  4. Equation with same base: Now we have an equation with the same base on both sides: 25x+8=2152^{5x+8} = 2^{-15}
  5. Set exponents equal: Since the bases are the same, we can set the exponents equal to each other: 5x+8=155x + 8 = -15
  6. Solve for x: Solve for x by subtracting 88 from both sides of the equation.5x+88=1585x + 8 - 8 = -15 - 85x=235x = -23
  7. Isolate x: Divide both sides by 55 to isolate x.\newline5x5=235\frac{5x}{5} = \frac{-23}{5}\newlinex=235x = \frac{-23}{5}

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