Question

topic: evaluate the expressions with rational exponents fill in the missing values of the table based on the growth that is described. 12. the growth in the table is triple at each whole year. years 0 ½ 1 ½ 2 ½ 3 ½ 4 bacteria 2 6 13. the growth in the table is triple at each whole year. years 0 ⅓ ⅔ 1 ⅘ ⅚ 2 ⅞ ⅘ bacteria 2 6 14. the values in the table grow by a factor of four at each whole year. years 0 ½ 1 ½ 2 ½ 3 ½ 4 bacteria 2 8 go topic: simplifying exponents simplify the following expressions using exponent rules and relationships, write your answers in exponential form. (for example: 2²·2⁵ = 2⁷) 15. 3²·3⁵ 16. 5³/5² 17. 2⁻⁵ 18. 17⁰ 19. 7⁵·7³/7²·7⁴ 20. 3⁻²·3⁵/3⁷

Explanation:

12.

Step1: Find the growth - rate formula

The growth is triple at each whole - year. The general formula for exponential growth is $y = a\cdot b^x$, where $a$ is the initial amount and $b$ is the growth factor. Here $a = 2$ and $b = 3$. For non - whole years, we use the property of exponents. If $x$ is the number of years, then $y=2\cdot3^x$.

Step2: Calculate values for $x=\frac{1}{2}$

When $x = \frac{1}{2}$, $y = 2\cdot3^{\frac{1}{2}}=2\sqrt{3}$. When $x=\frac{3}{2}$, $y = 2\cdot3^{\frac{3}{2}}=2\cdot3\cdot\sqrt{3}=6\sqrt{3}$. When $x = 2$, $y=2\cdot3^2 = 18$. When $x=\frac{5}{2}$, $y = 2\cdot3^{\frac{5}{2}}=2\cdot3^2\cdot\sqrt{3}=18\sqrt{3}$. When $x = 3$, $y=2\cdot3^3 = 54$. When $x=\frac{7}{2}$, $y = 2\cdot3^{\frac{7}{2}}=2\cdot3^3\cdot\sqrt{3}=54\sqrt{3}$. When $x = 4$, $y=2\cdot3^4 = 162$.

Step1: Determine the growth - rate formula

The growth factor $b = 3$ and the initial amount $a = 2$. The formula is $y = 2\cdot3^x$. For fractional years $x=\frac{n}{3}$, we calculate $y = 2\cdot3^{\frac{n}{3}}$.

Step2: Calculate values

When $x=\frac{1}{3}$, $y = 2\cdot3^{\frac{1}{3}}$. When $x=\frac{2}{3}$, $y = 2\cdot3^{\frac{2}{3}}$. When $x=\frac{4}{3}$, $y = 2\cdot3^{\frac{4}{3}}=2\cdot3\cdot3^{\frac{1}{3}} = 6\cdot3^{\frac{1}{3}}$. When $x=\frac{5}{3}$, $y = 2\cdot3^{\frac{5}{3}}=2\cdot3\cdot3^{\frac{2}{3}} = 6\cdot3^{\frac{2}{3}}$. When $x=\frac{7}{3}$, $y = 2\cdot3^{\frac{7}{3}}=2\cdot3^2\cdot3^{\frac{1}{3}} = 18\cdot3^{\frac{1}{3}}$. When $x=\frac{8}{3}$, $y = 2\cdot3^{\frac{8}{3}}=2\cdot3^2\cdot3^{\frac{2}{3}} = 18\cdot3^{\frac{2}{3}}$.

Step1: Identify the growth - rate formula

The initial amount $a = 2$ and the growth factor $b = 4$. The formula is $y = 2\cdot4^x$.

Step2: Calculate values for different $x$

When $x=\frac{1}{2}$, $y = 2\cdot4^{\frac{1}{2}}=2\cdot2 = 4$. When $x=\frac{3}{2}$, $y = 2\cdot4^{\frac{3}{2}}=2\cdot4\cdot\sqrt{4}=2\cdot4\cdot2 = 16$. When $x = 2$, $y=2\cdot4^2 = 32$. When $x=\frac{5}{2}$, $y = 2\cdot4^{\frac{5}{2}}=2\cdot4^2\cdot\sqrt{4}=2\cdot16\cdot2 = 64$. When $x = 3$, $y=2\cdot4^3 = 128$. When $x=\frac{7}{2}$, $y = 2\cdot4^{\frac{7}{2}}=2\cdot4^3\cdot\sqrt{4}=2\cdot64\cdot2 = 256$. When $x = 4$, $y=2\cdot4^4 = 512$.

Answer:

Years	0	$\frac{1}{2}$	1	$\frac{3}{2}$	2	$\frac{5}{2}$	3	$\frac{7}{2}$	4

13.