# Topic 1 Integer Exponents

## 1.1 Don’t Be Tricked

A pizza shop sales 12-inches pizza and 8-inches pizza at the price $12/each and $6/each respectively. With $12, would you like to order one 12-inches and two 8-inches. Why?

A sheet of everyday copy paper is about 0.01 millimeter thick. Repeat folding along a different side 20 times. Now, how thick do you think the folded paper is?

## 1.2 Properties of Exponents

For an integer \(n\), and an expression \(x\), the mathematical operation of the \(n\)-times repeated multiplication of \(x\) is call exponentiation, written as \(x^n\), that is, \[ x^n=\underbrace{x\cdot x \cdots x}_{n~\text{factors of}~x}. \]

In the notation \(x^n\), \(n\) is called ** the exponent**, \(x\) is called

**, and \(x^n\) is called**

*the base***, read as “\(x\) raised to the \(n\)-th power”, “\(x\) to the \(n\)-th power”, “\(x\) to the \(n\)-th”, “\(x\) to the power of \(n\)” or “\(x\) to the \(n\)”.**

*the power*Property | Example |
---|---|

The product rule \[x^m\cdot x^n=x^{m+n}.\] | \[2x^2\cdot (-3x^3)=-6x^5.\] |

The quotient rule (for \(x\neq 0\).) \[\dfrac{x^m}{x^n}= \begin{cases} x^{m-n} & \text{if} m\ge n.\\[1em] \dfrac{1}{x^{n-m}} & \text{if} m\le n. \end{cases} \] | \[\frac{15x^5}{5x^2}=3x^3;\] \[\frac{-3x^2}{6x^3}=-\frac{1}{2x}.\] |

The zero exponent rule (for \(x\neq 0\).) \[x^0=1.\] | \[(-2)^0=1;\] \[-x^0=-1.\] |

The negative exponent rule (for \(x\neq 0\).) \[x^{-n}=\dfrac{1}{x^n} \quad\text{and}\quad \dfrac{1}{x^{-n}}=x^n.\] | \[(-2)^{-3}=\frac{1}{(-2)^3}=-\frac18;\] \[\frac{x^{-2}}{x^{-3}}=\frac{x^3}{x^2}=x.\] |

The power to a power rule \[\left(x^a\right)^b=x^{ab}.\] | \[\left(2^{2}\right)^3=2^6=64;\] \[\left(x^2\right)^3=x^6.\] |

The product raised to a power rule \[(xy)^n=x^ny^n.\] | \[\left(-2x\right)^{2}=(-2)^2x^2=4x^2.\] |

The quotient raised to a power rule (for \(y\neq 0\).) \[\left(\dfrac{x}{y}\right)^n=\dfrac{x^n}{y^n}.\] | \[ \left(\dfrac{x}{-2}\right)^{3}=\dfrac{x^3}{(-2)^3}=-\dfrac{x^3}{8}.\] |

**Order of Basic Mathematical Operations**

In mathematics, the order of operations reflects conventions about which procedure should be performed first. There are four levels (from the highest to the lowest):

**Parenthesis**; **Exponentiation**; **Multiplication and Division**; **Addition and Subtraction**.

Within the same level, the convention is to perform from the left to the right.

**Example 1.1 **
Simplify. **Write with positive exponents.**
\[
\left(\dfrac{2y^{-2}z^{-5}}{4x^{-3}y^6}\right)^{-4}.
\]

*Solution. *
The idea is to simplify the base first and rewrite using positive exponents only.

\[ \begin{aligned} \left(\dfrac{2y^{-2}z^{-5}}{4x^{-3}y^6}\right)^{-4} =&\left(\dfrac{x^3}{2z^{5}y^8}\right)^{-4}\\ =&\left(\dfrac{2z^{5}y^8}{x^3}\right)^4\\ =&\dfrac{2^4(z^{5})^4(y^8)^4}{(x^3)^4}\\ =&\dfrac{16y^{32}z^{20}}{x^{12}}.\\ \end{aligned} \]

**Simplify (at least partially) the problem first**

To avoid mistakes when working with negative exponents, it’s better to apply the negative exponent rule to change negative exponents to positive exponents and simplify the base first.

## 1.3 Practice

**Problem 1.1 **
Simplify. **Write with positive exponents.**

- \((3a^2b^3c^2)(4abc^2)(2b^2c^3)\)
- \(\dfrac{4y^3z^0}{x^2y^2}\)
- \((-2)^{-3}\)

**Problem 1.2 **
Simplify. **Write with positive exponents.**

- \(\dfrac{-u^0v^{15}}{v^{16}}\)
- \((-2a^3b^2c^0)^3\)
- \(\dfrac{m^5 n^{2}}{(mn)^3}\)

**Problem 1.3 **
Simplify. **Write with positive exponents.**

- \((-3a^2x^3)^{-2}\)
- \(\left(\dfrac{-x^0y^3}{2wz^2}\right)^3\)
- \(\dfrac{3^{-2}a^{-3}b^5}{x^{-3}y^{-4}}\)

**Problem 1.4 **
Simplify. **Write with positive exponents.**

- \(\left(-x^{-1}(-y)^2\right)^3\)
- \(\left(\dfrac{6x^{-2}y^5}{2y^{-3}z^{-11}}\right)^{-3}\)
- \(\dfrac{(3 x^{2} y^{-1})^{-3}(2 x^{-3} y^{2})^{-1}}{(x^{6} y^{-5})^{-2}}\)

**Problem 1.5 **
A store has large size and small size watermelons. A large one cost $4 and a small one $1. Putting on the same table, a smaller watermelons has only half the height of the larger one. Given $4, will you buy a large watermelon or 4 smaller ones? Why?