Question

1 which expression equals $2^7$?
a $2 + 2 + 2 + 2 + 2 + 2 + 2$
b $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
c $2 \cdot 7$
d $2 + 7$
2 evaluate the expression $3 \cdot 5^x$ when $x$ is 2.
3 the graph shows the yearly balance, in dollars, in an investment account.
graph with x-axis (number of years) from 0 to 14, y-axis (amount in dollars) from 0 to 5000, points plotted starting at (0,1000) and increasing
a. what is the initial balance in the account?
b. is the account growing by the same number of dollars each year? explain how you know.
c. a second investment account starts with $2,000 and grows by $150 each year. sketch the values of this account on the graph.
d. how does the growth of balances in the two account balances compare?

Explanation:

Step1: Recognize exponent definition

$2^7$ means 2 multiplied 7 times.

Step2: Match to correct option

Option B is $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, which matches.
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Step1: Substitute $x=2$ into expression

$3 \cdot 5^x = 3 \cdot 5^2$

Step2: Calculate the power first

$5^2 = 25$

Step3: Multiply by 3

$3 \cdot 25 = 75$
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Part a: Identify initial balance

Initial balance is at $x=0$ (0 years), which is the y-intercept.

Part b: Check linear growth

Linear growth has equal vertical differences between consecutive points. The graph curves upward, so differences increase.

Part c: Define second account's formula

The balance $y = 2000 + 150x$, where $x$ is years. Calculate points:
$x=0$: $y=2000$; $x=2$: $y=2300$; $x=4$: $y=2600$; $x=6$: $y=2900$; $x=8$: $y=3200$; $x=10$: $y=3500$; $x=12$: $y=3800$; $x=14$: $y=4100$. Plot these as a straight line.

Part d: Compare growth types

First account has exponential (increasing rate) growth; second has linear (constant rate) growth.

Answer:

1. B. $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
2. 75

3.
a. $\$1000$
b. No. The graph is curved, so the annual increase in balance gets larger each year, not the same dollar amount.
c. (Plot the straight line through points (0,2000), (2,2300), (4,2600), (6,2900), (8,3200), (10,3500), (12,3800), (14,4100))
d. The first investment account grows exponentially (its balance increases by a larger dollar amount each year, accelerating over time), while the second account grows linearly (its balance increases by a constant $\$150$ each year at a steady, unchanging rate).