Topic 1 Integer Exponents

1.1 Don’t Be Tricked

  1. 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?

  2. 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 the base, and \(x^n\) is called the power, 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 \(n\)”.

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.

  1. \((3a^2b^3c^2)(4abc^2)(2b^2c^3)\)
  2. \(\dfrac{4y^3z^0}{x^2y^2}\)
  3. \((-2)^{-3}\)

Problem 1.2 Simplify. Write with positive exponents.

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

Problem 1.3 Simplify. Write with positive exponents.

  1. \((-3a^2x^3)^{-2}\)
  2. \(\left(\dfrac{-x^0y^3}{2wz^2}\right)^3\)
  3. \(\dfrac{3^{-2}a^{-3}b^5}{x^{-3}y^{-4}}\)

Problem 1.4 Simplify. Write with positive exponents.

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