Question

1. what is the width of the rectangle written as an exponential expression? 2^(-1) m area = 2^(-4) m^2 7 m 30. which expressions are equivalent to 2^(-11)? select all that apply. 2^(-23)/2^(-12) 2^(-7) 2^(-4) 2^2/2^(-2) 2^2 2^9 1/2^11

Explanation:

Step1: Recall area formula for rectangle

The area formula of a rectangle is $A = l\times w$, where $A$ is the area, $l$ is the length and $w$ is the width. Given $A = 2^{-4}\text{ m}^2$ and $l=2^{-1}\text{ m}$. We need to find $w$. So, $w=\frac{A}{l}$.

Step2: Substitute values and use exponent - rule

Substitute $A = 2^{-4}$ and $l = 2^{-1}$ into the formula $w=\frac{A}{l}$, we get $w=\frac{2^{-4}}{2^{-1}}$. According to the rule $\frac{a^m}{a^n}=a^{m - n}$, then $w=2^{-4-(-1)}=2^{-4 + 1}=2^{-3}\text{ m}$.

For the second part:

Step1: Simplify each option using exponent - rules

Option 1: $\frac{2^{-23}}{2^{-12}}$

Using the rule $\frac{a^m}{a^n}=a^{m - n}$, we have $\frac{2^{-23}}{2^{-12}}=2^{-23-(-12)}=2^{-23 + 12}=2^{-11}$.

Option 2: $2^{-7}\cdot2^{-4}$

Using the rule $a^m\cdot a^n=a^{m + n}$, we get $2^{-7}\cdot2^{-4}=2^{-7+( - 4)}=2^{-11}$.

Option 3: $\frac{2^2}{2^{-2}}$

Using the rule $\frac{a^m}{a^n}=a^{m - n}$, we have $\frac{2^2}{2^{-2}}=2^{2-(-2)}=2^{2 + 2}=2^4
eq2^{-11}$.

Option 4: $2^2\cdot2^9$

Using the rule $a^m\cdot a^n=a^{m + n}$, we get $2^2\cdot2^9=2^{2 + 9}=2^{11}
eq2^{-11}$.

Option 5: $\frac{1}{2^{11}}$

Since $\frac{1}{a^n}=a^{-n}$, then $\frac{1}{2^{11}}=2^{-11}$.

Answer:

1. $2^{-3}\text{ m}$
2. $\frac{2^{-23}}{2^{-12}}$, $2^{-7}\cdot2^{-4}$, $\frac{1}{2^{11}}$