7. evaluate $sqrt3{-512}$
*8. find the percentage of increase or decrease from an original price of $35 to a new price of $49
*9. multiple choice 78 changes to 88. find the percent of increase or decrease.
a 10% b 12% c 13% d 11%
*10. analyze is it possible to have a percent of increase of more than 100%? is it possible to have a percent of decrease of more than 100%? explain.

Question

7. evaluate $sqrt3{-512}$
*8. find the percentage of increase or decrease from an original price of $35 to a new price of $49
*9. multiple choice 78 changes to 88. find the percent of increase or decrease.
a 10%  b 12%  c 13%  d 11%
*10. analyze is it possible to have a percent of increase of more than 100%? is it possible to have a percent of decrease of more than 100%? explain.

Accepted Answer

### Question 7: Evaluate $\sqrt[3]{-512}$
# Explanation:
## Step 1: Recall the cube root property
We know that for any real number $ a $, $\sqrt[3]{-a}=-\sqrt[3]{a}$. So we can rewrite $\sqrt[3]{-512}$ as $-\sqrt[3]{512}$.

## Step 2: Find the cube root of 512
We need to find a number $ x $ such that $ x^3 = 512 $. We know that $ 8\times8\times8 = 8^3=512 $. So $\sqrt[3]{512} = 8$.

## Step 3: Combine the results
Since $\sqrt[3]{-512}=-\sqrt[3]{512}$ and $\sqrt[3]{512} = 8$, then $\sqrt[3]{-512}=-8$.

# Answer: $-8$

### Question 8: Find the percentage of increase or decrease from an original price of \$35 to a new price of \$49
# Explanation:
## Step 1: Determine if it's an increase or decrease
Since the new price (\$49) is greater than the original price (\$35), it's a percentage increase.

## Step 2: Calculate the amount of increase
The amount of increase is $ \text{New Price}-\text{Original Price} = 49 - 35=\$14 $.

## Step 3: Use the percentage increase formula
The formula for percentage increase is $ \text{Percentage Increase}=\frac{\text{Amount of Increase}}{\text{Original Price}}\times100\% $. Substituting the values, we get $ \frac{14}{35}\times100\% $.

## Step 4: Simplify the fraction
$ \frac{14}{35}=\frac{2}{5} $ (dividing numerator and denominator by 7). Then $ \frac{2}{5}\times100\% = 40\% $.

# Answer: The percentage increase is $ 40\% $

### Question 9: Multiple Choice: 78 changes to 88. Find the percent of increase or decrease. (Options: A. 10%, B. 12%, C. 13%, D. 11%)
# Explanation:
## Step 1: Determine if it's an increase or decrease
Since 88 > 78, it's a percentage increase.

## Step 2: Calculate the amount of increase
The amount of increase is $ 88 - 78 = 10 $.

## Step 3: Use the percentage increase formula
The formula for percentage increase is $ \text{Percentage Increase}=\frac{\text{Amount of Increase}}{\text{Original Amount}}\times100\% $. Here, the original amount is 78, so we have $ \frac{10}{78}\times100\% \approx 12.82\% \approx 13\% $? Wait, no, wait: Wait, 88 - 78 = 10? Wait, 88 - 78 = 10? Wait, 78 + 10 = 88? No, 78+10=88? 78+10=88? 78+10=88, yes. Wait, but $ \frac{10}{78}\times100\% \approx 12.82\% $, which is approximately 13%? Wait, no, the options are A.10%, B.12%, C.13%, D.11%. Wait, maybe I made a mistake. Wait, 78 to 88: 88 - 78 = 10. Then $ \frac{10}{78}\times100=\frac{1000}{78}\approx12.82\%\approx13\% $? But let's check again. Wait, maybe the original number is 78, new is 88. So $ \frac{88 - 78}{78}\times100=\frac{10}{78}\times100\approx12.82\%$, which is closest to 13%? But wait, maybe I miscalculated. Wait, 78 * 1.12 = 78 + 78*0.12 = 78 + 9.36 = 87.36, 78*1.13=78+78*0.13=78+10.14=88.14, which is close to 88. So the closest is 13%? Wait, but the options have C as 13%. Wait, but let's do the calculation properly. $ \frac{10}{78}=\frac{5}{39}\approx0.1282 $, so 12.82% which is approximately 13% (option C) or 12%? Wait, 12% of 78 is 0.12*78=9.36, 78+9.36=87.36. 13% of 78 is 10.14, 78+10.14=88.14. The difference between 88 and 78 is 10, which is closer to 10.14 (13%) than 9.36 (12%). So the answer is C.

# Answer: C. 13%

### Question 10: Analyze: Is it possible to have a percent of increase of more than 100%? Is it possible to have a percent of decrease of more than 100%? Explain.
# Brief Explanations:
- **Percent of increase more than 100%**: Yes. For example, if the original amount is 5 and the new amount is 15, the increase is $ 15 - 5 = 10 $. The percentage increase is $ \frac{10}{5}\times100\% = 200\% $, which is more than 100%. This is because the amount of increase can be more than the original amount (e.g., doubling, tripling, etc.).
- **Percent of decrease more than 100%**: No. The maximum possible decrease occurs when the new amount becomes 0 (since you can't have a negative amount in most real - world contexts like price, quantity). If the original amount is $ a $ ( $ a>0 $ ), the amount of decrease is $ a - 0=a $. The percentage decrease is $ \frac{a}{a}\times100\% = 100\% $. So you can't have a percent decrease of more than 100% because you can't decrease by more than the original amount (as the new amount can't be negative in typical scenarios).

# Answer: It is possible to have a percent of increase of more than 100% (e.g., when the new amount is more than double the original amount). It is not possible to have a percent of decrease of more than 100% because the maximum decrease is when the new amount is 0, resulting in a 100% decrease.

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QUESTION IMAGE

Question

Explanation:

Question 7: Evaluate $\sqrt[3]{-512}$

Step 1: Recall the cube root property

Step 2: Find the cube root of 512

Step 3: Combine the results

Step 1: Determine if it's an increase or decrease

Step 2: Calculate the amount of increase

Step 3: Use the percentage increase formula

Step 4: Simplify the fraction

Step 1: Determine if it's an increase or decrease

Step 2: Calculate the amount of increase

Step 3: Use the percentage increase formula

Answer:

Question 8: Find the percentage of increase or decrease from an original price of \$35 to a new price of \$49