Question

a student has a coin with sides heads, h, and tails, t, and a stack of three cards labeled with the numbers 1, 2, and 3. the student flips the coin once and picks one card at random.
make an organized list of the sample space
(h,1) (h,2) (h,3)
(t,1) (t,2) (t,3)

simplify the expression
$(4^{3})^{2} \cdot 4^{-8}$

tuesday
select the best estimate for each number.

	$5 \times 10^{-5}$	$5 \times 10^{4}$	$5 \times 10^{-3}$	$5 \times 10^{6}$
0.005376	$\square$	$\square$	$\square$	$\square$
4,632,125	$\square$	$\square$	$\square$	$\square$
0.0000492	$\square$	$\square$	$\square$	$\square$

find $(9.3 \times 10^{6}) + (1.8 \times 10^{4})$.
express your answer in scientific notation.

Explanation:

1. Make an Organized List of the Sample Space

Step1: List coin outcomes

The coin has 2 outcomes: \( H \) (heads) and \( T \) (tails).

Step2: Pair with card numbers

For each coin outcome, pair with card numbers 1, 2, 3:

• With \( H \): \( (H,1) \), \( (H,2) \), \( (H,3) \)
• With \( T \): \( (T,1) \), \( (T,2) \), \( (T,3) \)

Step1: Apply power of a power

Use \( (a^m)^n = a^{m \cdot n} \):
\( (4^3)^2 = 4^{3 \cdot 2} = 4^6 \)

Step2: Apply product of powers

Use \( a^m \cdot a^n = a^{m + n} \):
\( 4^6 \cdot 4^{-8} = 4^{6 + (-8)} = 4^{-2} \)

Step3: Simplify negative exponent

Use \( a^{-n} = \frac{1}{a^n} \):
\( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \)

Step1: Rewrite to same exponent

Rewrite \( 1.8 \times 10^4 \) as \( 0.018 \times 10^6 \) (since \( 10^4 = 10^{-2} \times 10^6 \)).

Step2: Add coefficients

\( 9.3 + 0.018 = 9.318 \)

Step3: Combine with \( 10^6 \)

\( 9.318 \times 10^6 \)

Answer:

\( (H,1) \), \( (H,2) \), \( (H,3) \), \( (T,1) \), \( (T,2) \), \( (T,3) \)

2. Simplify \( (4^3)^2 \cdot 4^{-8} \)