Question

1. use exponent rules to write each expression as a single power of 2. find the value of the expression. record these in the table. the first row is done for you. discuss with your partner any patterns you notice that show a relationship between the middle and right columns.

expression	power of 2	value
\\(\frac{2^5}{2^2}\\)
\\(\frac{2^5}{2^3}\\)
\\(\frac{2^5}{2^4}\\)
\\(\frac{2^5}{2^5}\\)
\\(\frac{2^5}{2^6}\\)	\\(2^{-1}\\)
\\(\frac{2^5}{2^7}\\)

1. what is the value of \\(5^0\\)?
2. what is the value of \\(3^{-1}\\)?
3. what is the value of \\(7^{-3}\\)?

Explanation:

Step1: Apply quotient of powers rule

For $\frac{2^a}{2^b}=2^{a-b}$

Row 2: $\frac{2^5}{2^2}=2^{5-2}=2^3$

Value: $2^3=8$

Row 3: $\frac{2^5}{2^3}=2^{5-3}=2^2$

Value: $2^2=4$

Row 4: $\frac{2^5}{2^4}=2^{5-4}=2^1$

Value: $2^1=2$

Row 5: $\frac{2^5}{2^5}=2^{5-5}=2^0$

Value: $2^0=1$

Row 6: $\frac{2^5}{2^6}=2^{5-6}=2^{-1}$

Value: $2^{-1}=\frac{1}{2}$

Row 7: $\frac{2^5}{2^7}=2^{5-7}=2^{-2}$

Value: $2^{-2}=\frac{1}{2^2}=\frac{1}{4}$

Step2: Evaluate zero exponent

Rule: $a^0=1$ for $a
eq0$
$5^0=1$

Step3: Evaluate negative exponent

Rule: $a^{-n}=\frac{1}{a^n}$
$3^{-1}=\frac{1}{3^1}=\frac{1}{3}$

Step4: Evaluate negative exponent

Rule: $a^{-n}=\frac{1}{a^n}$
$7^{-3}=\frac{1}{7^3}=\frac{1}{343}$

Answer:

1. Completed table:

expression	power of 2	value
$\frac{2^5}{2^2}$	$2^3$	8
$\frac{2^5}{2^3}$	$2^2$	4
$\frac{2^5}{2^4}$	$2^1$	2
$\frac{2^5}{2^5}$	$2^0$	1
$\frac{2^5}{2^6}$	$2^{-1}$	$\frac{1}{2}$
$\frac{2^5}{2^7}$	$2^{-2}$	$\frac{1}{4}$

Pattern: As the denominator's exponent increases by 1, the power of 2 decreases by 1, and the value is halved (or follows $2^{5-b}$ where $b$ is the denominator's exponent).

1. $1$
2. $\frac{1}{3}$
3. $\frac{1}{343}$