Question

PLEASE HELP ASAP DUE TONIGHT NEED YOUR HELP WILL MARK THE BRAINLEST

Question 1) Solve and show your work for each question.
NOTE: Remember per the rule of 9s if you have a repeating and non-repeating component you need to handle them separately.

(a)What is 0.42 expressed as a fraction in simplest form?
(b)What is 0.42 with a line on top of the 2 expressed as a fraction in simplest form?
(c)What is 0.42 with a line on top of the 2 and 4 expressed as a fraction in simplest form?

Question 2) Jacob estimates the square root of 75 in the following way:

√75=3√25=3×5=15

(a)Explain why his reasoning is incorrect.
(b)Estimate the square root of 75 to the nearest thousandths without using your calculator. Show your work.
NOTE: If you use a calculator you will get the wrong answer.


Question 3) Given the following numbers:
π,2√5,√5,√3

(a)Without using your calculator, approximate the decimal equivalent of each number to the nearest tenth. Make sure to show your work.

NOTE: If you use a calculator you will get the wrong answer.

(b)Order the numbers from least to greatest.

Answer 1

1a.

$0.42 = \frac{0.42}{1} = \frac{42}{100} = \frac{21}{50}$

1b. Manipulate the numbers using algebra.

$100x = 42.2222. \\ 10x = 4.2222...$

Then, subtract and solve for x.

$90x = 38 \\ x = \frac{38}{90} \\ x = \frac{19}{45}$

1c. Same thing, manipulate the numbers.

$100x = 42.4242... \\ x = 0.4242...$

Again, subtract and solve for x.

$99x = 42 \\ x = \frac{42}{99} \\ x = \frac{14}{33}$

2a. Jacob is pulling the 3 out of the root operation without getting the root of it.

$\sqrt{75} = \sqrt{3} \times \sqrt{25 } = 5 \sqrt{3}$

2b. Without using a calculator, the only way that is feasible is by trial and error, using perfect squares. For example: you know

$\sqrt{3}$

is somewhere between 1 and 2, as these are the closest perfect square roots to it. Try a value, say 1.5.

${1.5}^{2} = 2.25$

Not bad, seems a bit low. Lets try 1.6.

${1.6}^{2} = 2.56$

Seems like we can go bigger. 1.7 this time.

${1.7}^{2} = 2.89$

Alright, we are getting close to 3. 1.8.

${1.8}^{2} = 3.24$

Oops, too big. But no problem, because we now know that the value we are looking for is between 1.7 and 1.8, meaning the first decimal place must be 7. Let's keep going.

${1.75}^{2} = 3.0625 \\ {1.74}^{2} = 3.0276 \\ {1.73}^{2} = 2.9929$

Alright, the value is between 1.73 and 1.74.

${1.735}^{2} = 3.01... \\ {1.734}^{2} = 3.006... \\ {1.733}^{2} = 3.003... \\ {1.732}^{2} = 2.999...$

Bingo, your approximation is 1.732.

3. Convert each number to a tenth-rounded decimal.

$\pi = 3.1$

$\sqrt{3} = 1.7$

Find the square root of 5 first, as then you only have to double it for the remaining number.

${2.5}^{2} = 6.25 \\ {2.3}^{2} = 5.29 \\ {2.2}^{2} = 4.84 \\ {2.25}^{2} = 5.0625 \\ \sqrt{5} = 2.2$

The reason I take the 2.5 is to see whether to round up or down.

Then the last one is double the previous.

$2 \sqrt{5} = 4.4$

3b.

$\sqrt{3} < \sqrt{5} < \pi< 2 \sqrt{5}$

Hope this all helps.