Question

1. find the area of the rectangle. write the area as a single base raised to an exponent.

10^5 ft
10^2 ft
for problems 17 - 20, indicate whether each result is a positive or a negative number.

1. (-3)^2
2. (-4)^3
3. (-5)^(odd number)
4. (-6)^(even number)

Explanation:

Step1: Recall area formula for rectangle

The area formula of a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. Here $l = 10^{5}$ ft and $w=10^{2}$ ft.

Step2: Use exponent - multiplication rule

According to the rule $a^{m}\times a^{n}=a^{m + n}$, when $a = 10$, $m = 5$ and $n = 2$, we have $10^{5}\times10^{2}=10^{5 + 2}$.

Step3: Calculate the exponent sum

$5+2=7$, so $10^{5}\times10^{2}=10^{7}$ square - feet.

For problems 17 - 20:

Step4: Analyze $(-3)^{2}$

$(-3)^{2}=(-3)\times(-3)=9$, which is positive.

Step5: Analyze $(-4)^{3}$

$(-4)^{3}=(-4)\times(-4)\times(-4)=-64$, which is negative.

Step6: Analyze $(-5)^{\text{odd number}}$

When a negative number is raised to an odd power, the result is negative. For example, if the odd number is 1, $(-5)^{1}=-5$; if the odd number is 3, $(-5)^{3}=(-5)\times(-5)\times(-5)=-125$.

Step7: Analyze $(-6)^{\text{even number}}$

When a negative number is raised to an even power, the result is positive. For example, $(-6)^{2}=(-6)\times(-6) = 36$; $(-6)^{4}=(-6)\times(-6)\times(-6)\times(-6)=1296$.

Answer:

1. The area of the rectangle is $10^{7}$ square - feet.
2. Positive
3. Negative
4. Negative
5. Positive