Use properties of exponents to write the function in the form $f (t) = k a^{t}$ , where $k$ is a constant.
${(\frac{1}{2})}^{1 - 2 t}$
${(\frac{1}{2})}^{1 - 2 t} =$

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Answer

$f (t) = 2^{2 t - 1}$ is the final answer.

Steps

Step 1 ：The given function is ${(\frac{1}{2})}^{1 - 2 t}$ .

Step 2 ：We can rewrite it in the form $f (t) = k a^{t}$ by using the properties of exponents.

Step 3 ：The properties of exponents state that $a^{m - n} = \frac{a^{m}}{a^{n}}$ .

Step 4 ：So, we can rewrite the given function as ${\frac{1}{2}}^{1} \cdot {\frac{1}{2}}^{- 2 t}$ .

Step 5 ：Then, we can simplify this to $\frac{1}{2} \cdot 2^{2 t}$ .

Step 6 ：Finally, we can rewrite this as $2^{2 t - 1}$ .

Step 7 ：The function in the form $f (t) = k a^{t}$ is $f (t) = 1 \cdot 2^{2 t - 1}$ .

Step 8 ： $f (t) = 2^{2 t - 1}$ is the final answer.