###### tags: `MATA02-2022`
# MATA02-2022: Graded Assignment 1
**DUE: Friday, January 28, 11:59pm**

Please submit the assignment on Crowdmark. Be sure that you upload the correct pages for each problem. Where applicable, you must show your work. Make sure your writing is legible; if we cannot read your writing, we will mark it as incorrect.

**Note**: If you use any resources outside of this class, or consult with friends/classmates/tutors, you must document those sources on each problem. Getting help or assistance is permitted for this assignment, so long as it is properly cited.

## Problem 1 [8pts]
Use the distributive property on the following:
1. $4(2x+5)$
2. $2(5 + 2x + y)$

## Problem 2 [8pts]
Factor the following expressions as much as possible:
1. $15a + 36 + 18q$
2. $60x + 132 + 48n$

## Problem 3 [16pts]
Solve for $x$ in the following:
1. $4x - 7 = 12x -31$
2. $5x - 17 = 3x + 2$
3. $\frac{2}{3}x + \frac{5}{7} = \frac{1}{2}x + \frac{22}{21}$
4. $\frac{3}{7}x + \frac{2}{35} = \frac{5}{3}x - \frac{14}{15}$

## Problem 4 [8pts]
Write the following numbers as the sum of powers of 10.
For example $567 = 5 \times 10^2 + 6 \times 10^1 + 7 \times 10^0$.

1. 2549
2. 3791806

## Problem 5 [15pts]
In the following problems use the properties of exponents to write the given expression in the form $\frac{p}{q}$ where $p$ and $q$ are integers.

1. $\frac{3^{-4}}{5^{-4}}$
2. $25^{-\frac{1}{2}} - 625^{\frac{1}{4}}$
3. $\frac{2^{-2}3^3 5^4}{3^5 5^2 2^2}$

## Problem 6 [15pts]
Assume that all variables represent positive real numbers only. Write each of the following as a product or quotient of powers in which each variable occurs only once, and all exponents are positive.

1. $(x^3 y^{-2})^{-2}$
2. $(x^{\frac{1}{3}})^{-3} (x^{-3})^{-\frac{1}{3}}$
3. $\left( \frac{a^{-3}b^2 c^{-1}}{b^{-5} c^2 a^2} \right)^2$

## Problem 7 [15pts]
Find the least common multiple of the following pair of numbers.
1. 42, 75
2. 210, 693
3. 236, 578

## Problem 8 [15pts]
In class, we learned a little about the history of numbers, in addition to how we might reinvent them. For example, multiplication and fractions were invented by the Babylonians thousands of years before negative numbers. In our telling, you might invent a new type of number because otherwise some operation would be undefined. For example, we invented negative numbers so that subtraction always works, and we invented fractions so that division always works.

For this problem:
1. Describe a kind of number you would like to invent, and give it a name.
2. Argue why we might need that kind of number.
3. Then, describe the steps you'd have to do to make sure that the new kind of number works with all the old ones we already have.

**Note:** Be creative, and try to think like a mathematician. It's all right if you describe a new kind of number and it turns out to not work; I don't expect you to work through all the rigorous details. Indeed, most possible new kinds of numbers suffer from inconsistencies or aren't new numbers at all when you carefully mathematically analyze them; that's the reason we have so few types of numbers. So long as you have good responses to the three items above, you will get full credit.