Question

practice problems
1 which expression is equal to $4^0 \cdot 4^2$?
a 0
b 1
c 16
d 64
2 from unit 6, lesson 1
select all expressions that are equivalent to $3^8$.
a $8^3$
b $\frac{3^6}{3^2}$
c $3 \cdot 8$
d $(3^4)^2$
e $(3 \cdot 3)^4$
f $\frac{1}{3^8}$
3 a bee population is measured each week, and the results
are plotted on the graph.
a. what is the bee population when it is first measured?
b. is the bee population growing by the same factor
each week?
explain how you know.
c. what is an equation that models the bee population, $b$,
w weeks after it is first measured?

Explanation:

Problem 1

Step1: Recall exponent rule \(a^m \cdot a^n = a^{m + n}\)

\(4^0 \cdot 4^2 = 4^{0 + 2}\)

Step2: Simplify the exponent

\(4^{0 + 2}=4^2\)

Step3: Calculate \(4^2\)

\(4^2 = 16\)

We analyze each option using exponent rules (\(a^m \cdot a^n=a^{m + n}\), \(\frac{a^m}{a^n}=a^{m - n}\), \((a^m)^n=a^{m\cdot n}\)):

• Option A: \(8^3\) is not equal to \(3^8\) (different bases).
• Option B: \(\frac{3^6}{3^2}=3^{6 - 2}=3^4

eq3^8\)

• Option C: \(3\cdot8 = 24

eq3^8\)

• Option D: \((3^4)^2=3^{4\times2}=3^8\) (matches)
• Option E: \((3\cdot3)^4=(3^2)^4 = 3^{2\times4}=3^8\) (matches)
• Option F: \(\frac{1}{3^8}=3^{-8}

eq3^8\)

The first measurement is when \(w = 0\) (number of weeks). From the graph, when \(w = 0\), the bee population \(b\) is at the y - intercept. Looking at the graph, the y - intercept is 500 (since at \(w = 0\), the point is at 500 on the bee population axis).

Answer:

C. 16

Problem 2