Math 110 Course Resources - Exponential \& Logarithmic Functions Course Packet on solving for an unknown exponent Consider the equation $$ 2^{t^{2}-24}=(1 / 4)^{t} $$. Enter the number of solutions of the above equation. Determine the values of $$ t $$ that solve the above equation. Enter your answer as a comma-separated list of values. The order of the values does not matter. Enter DNE if there are no solutions.

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Solve the equation $$ 2^{t^{2}-24}=(1 / 4)^{t} $$.
Solve the equation by following steps:
- step0: Solve for $$t$$:
  $$2^{t^{2}-24}=\left(\frac{1}{4}ight)^{t}$$
- step1: Rewrite the expression:
  $$2^{t^{2}-24}=2^{-2t}$$
- step2: Set the exponents equal:
  $$t^{2}-24=-2t$$
- step3: Move the expression to the left side:
  $$t^{2}-24-\left(-2tight)=0$$
- step4: Remove the parentheses:
  $$t^{2}-24+2t=0$$
- step5: Factor the expression:
  $$\left(t-4ight)\left(t+6ight)=0$$
- step6: Separate into possible cases:
  $$\begin{align}&t-4=0\&t+6=0\end{align}$$
- step7: Solve the equation:
  $$\begin{align}&t=4\&t=-6\end{align}$$
- step8: Rewrite:
  $$t_{1}=-6,t_{2}=4$$
The equation $$2^{t^{2}-24}=(1 / 4)^{t}$$ has two solutions: $$t=-6$$ and $$t=4$$.
The equation has two solutions: $$t=-6$$ and $$t=4$$.

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