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$f(t)=1{,}000\left(\dfrac{1}{16}\right)^{^{\scriptsize \dfrac{t}{4}}}$ which of the following is an equivalent form of the function $f$ in which the base of the exponent is $\dfrac{1}{2}$ ?

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Transformations of functions

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Q.

f(t)=1{,}000\left(\dfrac{1}{16}\right)^{^{\scriptsize \dfrac{t}{4}}}

which of the following is an equivalent form of the function

f

in which the base of the exponent is

\dfrac{1}{2}

?

Express as Power of $1$ / $2$ : We need to rewrite the function $f(t) = 1,000 \times (1/16)^{(t/4)}$ with a base of $1/2$ for the exponent. To do this, we need to express $1/16$ as a power of $1/2$ .
Substitute in Function: Since $\frac{1}{16}$ is the same as $(\frac{1}{2})^4$ , we can substitute $(\frac{1}{2})^4$ in place of $\frac{1}{16}$ in the function $f(t)$ .
Rewrite with Substitution: Now, we rewrite the function $f(t)$ using this substitution: $f(t) = 1,000 \times \left(\left(\frac{1}{2}\right)^4\right)^{\frac{t}{4}}$ .
Apply Power Rule: Next, we apply the power of a power rule, which states that $(a^m)^n = a^{m*n}$ . In this case, we multiply the exponents $4$ and $t/4$ together.
Simplify Exponent: After applying the power of a power rule, we get $f(t) = 1,000 \times \left(\frac{1}{2}\right)^{4 \times \frac{t}{4}}$ .
Final Function: Simplify the exponent by multiplying $4$ and $\frac{t}{4}$ to get $t$ . The function now is $f(t) = 1,000 \times \left(\frac{1}{2}\right)^t$ .

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