###### tags: `MATA02-2022`
# Ungraded problems 2

You are to work on these questions only for the benefit of learning the material. These problems are meant for practice, not an assignment to be handed in.

## Exponents
If $a$ and $b$ are real numbers, $a > 0$, $b > 0$, and $m$ and $n$ are integer numbers, then

$$
a^m = \underbrace{a\cdot a \cdots a}_{m \text{ times}} \hspace{1em} \text{ and } \hspace{1em} a^{1/n} = \sqrt[n]{a}
$$

The following are the rules of exponents:
1. $a^0 = 1$
2. $(a^m)^n = a^{mn}$
3. $a^{m+n} = a^m a^n$
4. $a^{-m} = \frac{1}{a^m}$
5. $(ab)^m = a^m b^m$
6. $(a^m)^{\frac{1}{n}} = \left( a^{\frac{1}{n}} \right)^m = a^\frac{m}{n}$

**Examples**:

1. $2^{1/2} 2^{5/6} = 2^{1/2 + 5/6} = 2^{8/6} = 2^{4/3}$
2. $16^{1/4} 2^{-5} = (2)2^{-5} = 2^{-4} = \frac{1}{16}$
3. $(3^5 2^3)^2 = 3^{10} 2^6$
4. $81^{1/3} = (3^4)^{1/3} = 3^{4/3}$
5. $16^{-1/4} 3^2 = \frac{1}{16^{1/4}} 3^2 = \frac{3^2}{16^{1/4}}=\frac{9}{2}$

## Question 1:
In the following problems use the properties above to write the given expression in the "rational" form $p/q$ where $p$ and $q$ are integers.

*Note*: if $q=1$, you may simply write $p/1$ as $p$.

**Example**: $4^{\frac{1}{2}} + 4^{-\frac{1}{2}} = 4^{\frac{1}{2}} + \frac{1}{4^{\frac{1}{2}}} = 2 + \frac{1}{2} = \frac{5}{2}.$

1. $\frac{3^{-2}}{2^{-3}}$
2. $\frac{1}{2^{-1}}$
3. $\left( \frac{3}{5} \right)^{-1}$
4. $\left( -\frac{1}{3} \right)^{-2}$
5. $\frac{2^0}{3^{-2}}$
6. $\frac{5^{-1}}{3^{-2}}$
7. $(-8)^{-\frac{1}{3}}$
8. $(16)^{-\frac{1}{4}}$
9. $3^{-2} + 3$
10. $5^{-1} + 25^0$
11. $16^{-\frac{1}{2}} - 16^{\frac{1}{4}}$
12. $8^{-\frac{1}{3}} - 2^0$
13. $\frac{16^{\frac{1}{2}}}{8^{-\frac{2}{3}}}$
14. $4^{-1}+3^{-1}$
15. $\left(\frac{1}{5}\right)^{-1} - \left(\frac{1}{7}\right)^{-1}$

## Question 2:
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 but once, and all exponents are positive.

**Example**: $\left( \frac{x^{-1}y^2z^0}{x^3 y^{-4} z^2} \right)^{-1} = \frac{xy^{-2}z^0}{x^{-3}y^4 z^{-2}} = \frac{x^4 z^2}{y^6}$

1. $x^{-3}x^5$
2. $(x^2 y^{-3})^{-1}$
3. $\frac{x^5}{x^{-2}}$
4. $(x^{-3})^2$
5. $(x^\frac{1}{2})^{-3}$
6. $(x^3)^{-\frac{1}{3}}$
7. $(x^2 y^{-1})^{-\frac{1}{2}}$
8. $(x^3 y^{-2})^{-\frac{1}{6}}$
9. $(x^{-2}y^3)^0$
10. $\frac{x^{-1}}{y^{-1}}$
11. $\frac{x^{-2}}{y^{-3}}$
12. $\frac{a^2x^{-3}}{b^2 y^{-2}}$
13. $\frac{a^{-2} b^{-2} c}{ab^{-3}c^0}$
14. $\left(\frac{x^{-1}y^3}{2x^0 y^{-5}}\right)^{-2}$
15. $\left(\frac{a^{-1} b^{-2}}{3^0 ab}\right)^{-1}$

## Question 3:
In the following problems find the nonnegative integers $c$ and $r$ so that $a = b \times c + r$ where $0 \le r < b$.

You will have to perform division with remainder. Although you may use a calculator to help, I suggest that you do these practice problems by hand to get a better feel for what's happening.

**Example**: $a = 159, b =42$. Then divide 152 by 42 to get:
Hence $c = 3$ and $r = 33$

1. $a = 96, b=51$
2. $a = 397, b=73$
3. $a=1053, b=234$
4. $a=5639, b=132$

# Selected solutions
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<summary>Solutions to selected problems (click to expand)</summary>


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