Question

simplify completely. assume all variables represent positive values. (c^{2/5})^{3/7}
a) write the simplified expression using a positive exponent:
b) write the simplified expression as a radical expression:
*algebraic work must be shown to retain points on quizzes, midterm or final
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question 47
given that s(t)=12t - t^{2} is a position function where s(t) is measured in millimeters and t is time in seconds, find the average velocity (average rate of change) for s(t) between t = 1 and t = 3 seconds.
select an answer
*algebraic work must be shown to retain points on quizzes, midterm or final
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Explanation:

Step1: Simplify the exponent - part (a)

Use the power - of - a - power rule $(a^m)^n=a^{mn}$. Here $a = c$, $m=\frac{2}{5}$, and $n = \frac{3}{7}$. So $c^{\frac{2}{5}\times\frac{3}{7}}=c^{\frac{6}{35}}$.

Step2: Convert to radical form - part (b)

Recall that $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. For $a = c$, $m = 6$, and $n = 35$, we have $\sqrt[35]{c^{6}}$.

Step3: Calculate average velocity - Question 47

The average rate of change of a function $y = s(t)$ from $t=a$ to $t = b$ is given by $\frac{s(b)-s(a)}{b - a}$. First, find $s(1)$ and $s(3)$.
$s(1)=12\times1-1^{2}=12 - 1=11$.
$s(3)=12\times3-3^{2}=36 - 9 = 27$.
Then the average velocity $v=\frac{s(3)-s(1)}{3 - 1}=\frac{27 - 11}{2}=\frac{16}{2}=8$ mm/s.

Answer:

a) $c^{\frac{6}{35}}$
b) $\sqrt[35]{c^{6}}$
Question 47: 8