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Solve the exponential equation for x. {:[2^(8x-4)*16^(3x-5)=16^(9x+2)],[x=]:}

Solve the exponential equation for x x .\newline28x−4⋅163x−5=169x+2x=□ \begin{array}{l} 2^{8 x-4} \cdot 16^{3 x-5}=16^{9 x+2} \\ x=\square \end{array}

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Full solution

Q. Solve the exponential equation for x x .\newline28x−4⋅163x−5=169x+2x=□ \begin{array}{l} 2^{8 x-4} \cdot 16^{3 x-5}=16^{9 x+2} \\ x=\square \end{array}
  1. Recognize Power of 22: Recognize that 1616 is a power of 22, specifically 16=2416 = 2^4. This will allow us to rewrite the equation in terms of the same base.
  2. Rewrite Exponents: Rewrite 163x−516^{3x-5} and 169x+216^{9x+2} as (24)3x−5(2^4)^{3x-5} and (24)9x+2(2^4)^{9x+2} respectively, using the property abc=(ab)ca^{bc} = (a^b)^c.
  3. Simplify Exponents: Simplify the exponents on the right side of the equation by multiplying them out: (24)(3x−5)=2(4∗(3x−5))(2^4)^{(3x-5)} = 2^{(4*(3x-5))} and (24)(9x+2)=2(4∗(9x+2))(2^4)^{(9x+2)} = 2^{(4*(9x+2))}.
  4. Rewrite Equation: Now rewrite the equation with these simplifications: 28x−4×24×(3x−5)=24×(9x+2)2^{8x-4} \times 2^{4\times(3x-5)} = 2^{4\times(9x+2)}.
  5. Add Exponents: Add the exponents on the left side of the equation using the property am⋅an=am+na^m \cdot a^n = a^{m+n}: 28x−4⋅212x−20=28x−4+12x−202^{8x-4} \cdot 2^{12x-20} = 2^{8x-4 + 12x-20}.
  6. Combine Like Terms: Simplify the exponent on the left side by combining like terms: 28x−4+12x−20=220x−242^{8x-4 + 12x-20} = 2^{20x-24}.
  7. Set Exponents Equal: Now we have 220x−24=236x+82^{20x-24} = 2^{36x+8}. Since the bases are the same and the exponents must be equal for the equation to hold, we can set the exponents equal to each other: 20x−24=36x+820x-24 = 36x+8.
  8. Subtract 20x20x: Solve for xx by first subtracting 20x20x from both sides: −24=16x+8-24 = 16x + 8.
  9. Subtract 88: Now subtract 88 from both sides: −32=16x-32 = 16x.
  10. Divide by 1616: Finally, divide both sides by 1616 to solve for xx: x=−3216x = -\frac{32}{16}.
  11. Simplify Fraction: Simplify the fraction to get the final answer: x=−2x = -2.

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