f(t)=1,000((1)/(16))^((t)/(4)) Which of the following is an equivalent form of the function f in which the base of the exponent is (1)/(2) ? Choose 1 answer: (A) f(t)=125((1)/(2))^((t)/(4)) (B) f(t)=1,000((1)/(2))^(t) (C) f(t)=1,000((1)/(2))^(2t) (D) f(t)=1,000((1)/(2))^(4t)

Given Function Conversion: We are given the function f(t) = 1,000 * ((1)/(16))^((t)/(4)). We need to express this function with a base of (1)/(2) for the exponent. First, let's express (1)/(16) as a power of (1)/(2). (1)/(16) is the same as (1)/(2)^4 because (1)/(2)^4 = 1/16.Substitute Power of (1/2): Now, we can substitute (1)/(2)^4 back into the original function for (1)/(16). So, f(t) becomes f(t) = 1,000 * ((1)/(2)^4)^((t)/(4)).Apply Power Rule: Next, we apply the power of a power rule, which states that (a^m)^n = a^(m*n). Therefore, ((1)/(2)^4)^((t)/(4)) becomes (1)/(2)^(4*(t)/(4)).Simplify Exponent: Simplify the exponent by multiplying 4 by (t)/(4), which gives us (1)/(2)^(t). So, f(t) simplifies to f(t) = 1,000 * (1)/(2)^(t).Compare with Answer Choices: Now, we compare our simplified function to the answer choices. The correct answer choice that matches f(t) = 1,000 * (1)/(2)^(t) is (B) f(t) = 1,000 * (1)/(2)^(t).

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f(t)=1,000((1)/(16))^((t)/(4))
Which of the following is an equivalent form of the function
f in which the base of the exponent is
(1)/(2) ?
Choose 1 answer:
(A)

f(t)=125((1)/(2))^((t)/(4))
(B)
f(t)=1,000((1)/(2))^(t)
(C)
f(t)=1,000((1)/(2))^(2t)
(D)
f(t)=1,000((1)/(2))^(4t)

$f(t)=1,000\left(\frac{1}{16}\right)^{\frac{t}{4}}$ $\newline$ Which of the following is an equivalent form of the function $f$ in which the base of the exponent is $\frac{1}{2}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $f(t)=125\left(\frac{1}{2}\right)^{\frac{t}{4}}$ $\newline$ (B) $f(t)=1,000\left(\frac{1}{2}\right)^{t}$ $\newline$ (C) $f(t)=1,000\left(\frac{1}{2}\right)^{2 t}$ $\newline$ (D) $f(t)=1,000\left(\frac{1}{2}\right)^{4 t}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

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

\newline

Which of the following is an equivalent form of the function

f

in which the base of the exponent is

\frac{1}{2}

\newline

Choose

1

answer:

\newline

(A)

f(t)=125\left(\frac{1}{2}\right)^{\frac{t}{4}}

\newline

(B)

f(t)=1,000\left(\frac{1}{2}\right)^{t}

\newline

(C)

f(t)=1,000\left(\frac{1}{2}\right)^{2 t}

\newline

(D)

f(t)=1,000\left(\frac{1}{2}\right)^{4 t}

Given Function Conversion: We are given the function $f(t) = 1,000 \times \left(\frac{1}{16}\right)^{\frac{t}{4}}$ . We need to express this function with a base of $\frac{1}{2}$ for the exponent. $\newline$ First, let's express $\frac{1}{16}$ as a power of $\frac{1}{2}$ . $\newline$ $\frac{1}{16}$ is the same as $\frac{1}{2}^4$ because $\frac{1}{2}^4 = \frac{1}{16}$ .
Substitute Power of $(1/2)$ : Now, we can substitute $(1)/(2)^4$ back into the original function for $(1)/(16)$ . So, $f(t)$ becomes $f(t) = 1,000 \times ((1)/(2)^4)^{(t)/(4)}$ .
Apply Power Rule: Next, we apply the power of a power rule, which states that a^m)^n = a^{m*n}\. Therefore, \$\frac{1}{2}^4^{\frac{t}{ $4$ }} becomes $\frac{1}{2}$ ^{ $4$ *\frac{t}{ $4$ }}.
Simplify Exponent: Simplify the exponent by multiplying $4$ by $(t)/(4)$ , which gives us $(1)/(2)^{t}$ . So, $f(t)$ simplifies to $f(t) = 1,000 \times (1)/(2)^{t}$ .
Compare with Answer Choices: Now, we compare our simplified function to the answer choices. $\newline$ The correct answer choice that matches $f(t) = 1,000 \times \frac{1}{2^{t}}$ is (B) $f(t) = 1,000 \times \frac{1}{2^{t}}$ .