(d) Solve for $$ x $$ given $$ \left(\frac{1}{4}\right)^{x}=8 $$.

Question

UpStudy AI · Accepted Answer

1. Express the base $$ \frac{1}{4} $$ as a power of 2:
   $$
   \frac{1}{4} = 4^{-1} = (2^2)^{-1} = 2^{-2}.
   $$

2. Substitute into the equation:
   $$
   \left(2^{-2}\right)^{x} = 8.
   $$

3. Simplify the left side using the exponent rule $$ \left(a^{b}\right)^{c}=a^{bc} $$:
   $$
   2^{-2x} = 8.
   $$

4. Express 8 as a power of 2:
   $$
   8 = 2^3.
   $$

5. Now set the equation:
   $$
   2^{-2x} = 2^3.
   $$

6. Equate the exponents (since the bases are equal):
   $$
   -2x = 3.
   $$

7. Solve for $$ x $$:
   $$
   x = -\frac{3}{2}.
   $$

Thus, the solution is $$ x = -\frac{3}{2} $$.
$$ x = -\frac{3}{2} $$

(d) Solve for given .