\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
International Baccalaureate · DP 1: Analysis and Approaches SL

Exponents

⌚ ~434 min ▢ 52 blocks ✓ 221 exercises ➣ Prerequisites : Exponents

Comprehensive guide to exponents, covering positive, negative, and rational powers. Learn the 6 index laws, scientific notation, and the constant e. Master simplifying algebraic expressions and solving exponential equations, including quadratic forms and equating indices.

I Positive Exponents

Imagine you have a chessboard. You place two grains of wheat on the first square, four grains on the second square, eight grains on the third square, and so on, doubling the number of grains on each next square.
How many grains of wheat are on the last square of a chessboard with 64 squares?

Square number Number of grains
\(1\) \(2\)
\(2\) \(2 \times 2\)
\(3\) \(2 \times 2 \times 2\)
\(\vdots\) \(\vdots\)
\(64\) \(\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}\)
Rather than writing \(\overbrace{2 \times 2 \times \dots \times 2}^{64\ \text{factors}}\), we can write this product as \(2^{64}\).
This means there are \(2^{64}\) grains on the last square. Using a calculator:$$2^{64}=18\,446\,744\,073\,709\,551\,616.$$This is an enormous number!


Definition — Exponentiation
Exponentiation is repeated multiplication of a number by itself.
For a number \(a\) and a positive whole number \(n\),$$a^n = \overbrace{a \times a \times \dots \times a}^{n\ \text{factors}}.$$
Example
Write using exponent notation: \(5 \times 5 \times 5\).

\(5 \times 5 \times 5 = 5^3\)

Definition — Vocabulary
$$\begin{array}{|c|c|c|c|}\hline\text{Value} & \text{Expanded form} & \text{Exponent notation} & \text{Spoken form} \\ \hline2 & 2 & 2^1 & 2\ \text{or}\ 2\ \text{to the power of}\ 1 \\ 4 & 2 \times 2 & 2^2 & 2\ \text{squared or}\ 2\ \text{to the power of}\ 2 \\ 8 & 2 \times 2 \times 2 & 2^3 & 2\ \text{cubed or}\ 2\ \text{to the power of}\ 3 \\ 16 & 2 \times 2 \times 2 \times 2 & 2^4 & 2\ \text{to the power of}\ 4 \\ 32 & 2 \times 2 \times 2 \times 2 \times 2 & 2^5 & 2\ \text{to the power of}\ 5 \\ \hline\end{array}$$
Example
Find the value of \(2^3\).

$$\begin{aligned}[t]2^3 &= 2 \times 2 \times 2 \\ &= 8\end{aligned}$$

Skills to practice
  • Writing Repeated Multiplication in Exponent Form
  • Writing in Exponent Form from Verbal Expressions
  • Calculating Powers
  • Expressing Numbers in Exponent Form
  • Interpreting Powers
  • Evaluating Expressions with Powers
  • Checking Equality between Products and Powers
  • Writing Repeated Multiplication of an Algebraic Expression in Exponent Form
  • Writing Algebraic Expressions in Exponent Form from Verbal Descriptions
II Negative Exponents

To understand negative exponents, let's explore the pattern of multiplying by \(2\):
From this pattern, you can see:
  • \(2^1 = 2\)
  • \(2^2 = 2 \times 2\)
  • \(2^3 = 2 \times 2 \times 2\)
Because division is the inverse of multiplication, we can divide by \(2\) repeatedly to extend the pattern:
From this extended pattern, you can see:
  • \(2^0 = 1\)
  • \(2^{-1} = \dfrac{1}{2}\)
  • \(2^{-2} = \dfrac{1}{2 \times 2}\)
  • \(2^{-3} = \dfrac{1}{2 \times 2 \times 2}\)

Definition — Exponentiation for a negative exponent
For a non-zero number \(a\) and a positive integer \(n\), we extend exponentiation to negative exponents by:$$\begin{aligned}[t]a^{-n} &= \dfrac{1}{\underbrace{a \times a \times \dots \times a}_{n\ \text{factors}}} \\ &= \dfrac{1}{a^n}\end{aligned} \qquad \text{and} \qquad a^0 = 1 \quad (a \neq 0).$$In particular, \(a^{-1} = \dfrac{1}{a}\). A negative exponent means we take the reciprocal of the corresponding positive power.
Example
Write \(3^{-2}\) as a fraction.

$$\begin{aligned}[t]3^{-2} &= \dfrac{1}{3 \times 3} \\ &= \dfrac{1}{9}\end{aligned}$$

Skills to practice
  • Writing Negative Exponents as Fractions
  • Writing Fractions as Negative Exponents
III Rational Exponents

We know about positive exponents, like \(5^3 = 5 \times 5 \times 5\), and also about negative exponents, like \(5^{-3} = \dfrac{1}{5 \times 5 \times 5}\).
But what about fractional exponents?
Using the exponent laws, let's see what happens with \(\textcolor{colordef}{5^{\frac{1}{2}}}\):$$\begin{aligned}\textcolor{colordef}{5^{\frac{1}{2}}} \times \textcolor{colordef}{5^{\frac{1}{2}}} &= 5^{\frac{1}{2}+\frac{1}{2}} \\ &= 5^1 \\ &= \textcolor{olive}{5}\end{aligned}$$And by the definition of the square root:
$$\textcolor{colorprop}{\sqrt{5}} \times \textcolor{colorprop}{\sqrt{5}} = \textcolor{olive}{5}$$By comparing these two results, we see that:$$\textcolor{colordef}{5^{\frac{1}{2}}} \times \textcolor{colordef}{5^{\frac{1}{2}}} = \textcolor{colorprop}{\sqrt{5}} \times \textcolor{colorprop}{\sqrt{5}}$$So, we can deduce that:$$\textcolor{colordef}{5^{\frac{1}{2}}} = \textcolor{colorprop}{\sqrt{5}}$$In this chapter, we will only use rational exponents with positive bases, so that roots like \(\sqrt[n]{a}\) are real numbers. This shows us that we can use fractional exponents to represent roots, extending our understanding of exponents to include rational exponents.

Definition — Rational Exponent
For a positive number \(a\) and positive integers \(m\) and \(n\),$$\begin{aligned}a^{\frac{1}{2}} &= \sqrt{a}, \\ a^{\frac{1}{n}} &= \sqrt[n]{a}, \\ a^{\frac{m}{n}} &= \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m.\end{aligned}$$
Ex 1
Skills to practice
  • Expressing roots using exponents
  • Calculating Powers and Rounding
IV Exponent Law 1

Let's look at an example:$$\begin{aligned}\textcolor{colordef}{7}^{\textcolor{colorprop}{3}} \times \textcolor{colordef}{7}^{\textcolor{olive}{2}}&= \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{colorprop}{3}\,\text{factors}} \times \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{olive}{2}\,\text{factors}} \\ &= \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{colorprop}{3}+\textcolor{olive}{2}\,\text{factors}} \\ &= \textcolor{colordef}{7}^{\textcolor{colorprop}{3}+\textcolor{olive}{2}}.\end{aligned}$$In this example we are multiplying two powers with the same base (7).
We can see that we keep the base and add the exponents: \(3 + 2 = 5\).
In general, when a number \(\textcolor{colordef}{a}\) is raised to the power \(\textcolor{colorprop}{m}\) and multiplied by the same number raised to the power \(\textcolor{olive}{n}\), that is$$\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}},$$the result is equal to \(\textcolor{colordef}{a}\) raised to the sum of the exponents:$$\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}}.$$

Proposition — Exponent Law 1
When we multiply two powers with the same base, we keep the base and add the exponents:$$\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}}.$$

Let \(\textcolor{colordef}{a}\) be a number and let \(\textcolor{colorprop}{m}\) and \(\textcolor{olive}{n}\) be whole numbers.$$\begin{aligned}\textcolor{colordef}{a}^{\textcolor{colorprop}{m}} \times \textcolor{colordef}{a}^{\textcolor{olive}{n}}&= \overbrace{\textcolor{colordef}{a} \times \cdots \times \textcolor{colordef}{a}}^{\textcolor{colorprop}{m}\ \text{factors}} \times \overbrace{\textcolor{colordef}{a} \times \cdots \times \textcolor{colordef}{a}}^{\textcolor{olive}{n}\ \text{factors}} \\ &= \overbrace{\textcolor{colordef}{a} \times \cdots \times \textcolor{colordef}{a}}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}\ \text{factors}} \\ &= \textcolor{colordef}{a}^{\textcolor{colorprop}{m}+\textcolor{olive}{n}}.\end{aligned}$$

Example
Simplify \(5^2\times 5^4\).

$$\begin{aligned}\textcolor{colordef}{5}^{\textcolor{colorprop}{2}} \times \textcolor{colordef}{5}^{\textcolor{olive}{4}}&= \textcolor{colordef}{5}^{\textcolor{colorprop}{2}+\textcolor{olive}{4}} && \text{(same base, add exponents)} \\ &= \textcolor{colordef}{5}^{6}.\end{aligned}$$

Skills to practice
  • Simplifying Products of Powers
  • Simplifying Products of Algebraic Powers
  • Identifying Correct Exponential Expressions
  • Simplifying Expressions of Powers
V Exponent Law 2

Let's look at an example:$$\begin{aligned}\dfrac{\textcolor{colordef}{7}^{\textcolor{colorprop}{5}}}{\textcolor{colordef}{7}^{\textcolor{olive}{2}}}&= \dfrac{\overbrace{\cancel{\textcolor{colordef}{7}} \times \cancel{\textcolor{colordef}{7}} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{colorprop}{5}\,\text{factors}}} {\underbrace{\cancel{\textcolor{colordef}{7}} \times \cancel{\textcolor{colordef}{7}}}_{\textcolor{olive}{2}\,\text{factors}}}\\ &= \overbrace{\textcolor{colordef}{7} \times \textcolor{colordef}{7} \times \textcolor{colordef}{7}}^{\textcolor{colorprop}{5} - \textcolor{olive}{2}\,\text{factors}}\\ &= \textcolor{colordef}{7}^{\textcolor{colorprop}{5} - \textcolor{olive}{2}}\end{aligned}$$In general, when a number \(\textcolor{colordef}{a}\) is raised to the power \(\textcolor{colorprop}{m}\) and divided by the same number raised to the power \(\textcolor{olive}{n}\), that is$$\dfrac{\textcolor{colordef}{a}^{\textcolor{colorprop}{m}}}{\textcolor{colordef}{a}^{\textcolor{olive}{n}}},$$the result is \(\textcolor{colordef}{a}\) raised to the difference of the exponents:$$\dfrac{\textcolor{colordef}{a}^{\textcolor{colorprop}{m}}}{\textcolor{colordef}{a}^{\textcolor{olive}{n}}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m} - \textcolor{olive}{n}}.$$

Proposition — Exponent Law 2
For \(a\neq 0\) and any numbers \(m\) and \(n\),$$\dfrac{\textcolor{colordef}{a}^{\textcolor{colorprop}{m}}}{\textcolor{colordef}{a}^{\textcolor{olive}{n}}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m} - \textcolor{olive}{n}}$$
Example
Simplify \(\dfrac{5^7}{5^3}\).

$$\begin{aligned}\dfrac{\textcolor{colordef}{5}^{\textcolor{colorprop}{7}}}{\textcolor{colordef}{5}^{\textcolor{olive}{3}}}&= \textcolor{colordef}{5}^{\textcolor{colorprop}{7} - \textcolor{olive}{3}} \\ &= \textcolor{colordef}{5}^{4}\end{aligned}$$

Skills to practice
  • Simplifying Fractions of Powers
  • Simplifying Fractions of Algebraic Powers
VI Exponent Law 3

Let's look at an example:$$\begin{aligned}\left(\textcolor{colordef}{5}^{\textcolor{colorprop}{2}}\right)^{\textcolor{olive}{3}}&= (\overbrace{\textcolor{colordef}{5}\times \textcolor{colordef}{5}}^{\textcolor{colorprop}{2}\,\text{factors}})^{\textcolor{olive}{3}} \\ &= \overbrace{(\overbrace{\textcolor{colordef}{5}\times \textcolor{colordef}{5}}^{\textcolor{colorprop}{2}\,\text{factors}}) \times (\overbrace{\textcolor{colordef}{5}\times \textcolor{colordef}{5}}^{\textcolor{colorprop}{2}\,\text{factors}}) \times (\overbrace{\textcolor{colordef}{5}\times \textcolor{colordef}{5}}^{\textcolor{colorprop}{2}\,\text{factors}})}^{\textcolor{olive}{3}\,\text{factors}} \\ &= \textcolor{colordef}{5}^{\textcolor{colorprop}{2} + \textcolor{colorprop}{2} +\textcolor{colorprop}{2}}\\ &= \textcolor{colordef}{5}^{\textcolor{colorprop}{2} \times \textcolor{olive}{3}}\end{aligned}$$In general, when a number \(\textcolor{colordef}{a}\) is raised to the power \(\textcolor{colorprop}{m}\), and that result is raised to the power \(\textcolor{olive}{n}\), that is$$\left(\textcolor{colordef}{a}^{\textcolor{colorprop}{m}}\right)^{\textcolor{olive}{n}},$$the result is \(\textcolor{colordef}{a}\) raised to the product of the exponents:$$\left(\textcolor{colordef}{a}^{\textcolor{colorprop}{m}}\right)^{\textcolor{olive}{n}}= \textcolor{colordef}{a}^{\textcolor{colorprop}{m} \times \textcolor{olive}{n}}.$$

Proposition — Exponent Law 3
For \(a\neq 0\) and any numbers \(m\) and \(n\),$$\left(\textcolor{colordef}{a}^{\textcolor{colorprop}{m}}\right)^{\textcolor{olive}{n}} = \textcolor{colordef}{a}^{\textcolor{colorprop}{m} \times \textcolor{olive}{n}}$$
Example
Simplify \(\left(\textcolor{colordef}{5}^{\textcolor{colorprop}{2}}\right)^{\textcolor{olive}{5}}\).

$$\begin{aligned}[t]\left(\textcolor{colordef}{5}^{\textcolor{colorprop}{2}}\right)^{\textcolor{olive}{5}}&= \textcolor{colordef}{5}^{\textcolor{colorprop}{2} \times \textcolor{olive}{5}} \\ &= \textcolor{colordef}{5}^{10}\end{aligned}$$

Skills to practice
  • Simplifying Powers of Powers
  • Simplifying Powers of Powers
VII Exponent Law 4

Let's look at an example:$$\begin{aligned}(\textcolor{colordef}{3} \times \textcolor{colorprop}{5})^{\textcolor{olive}{2}}&= (\textcolor{colordef}{3} \times \textcolor{colorprop}{5}) \times (\textcolor{colordef}{3} \times \textcolor{colorprop}{5}) \\ &= \textcolor{colordef}{3} \times \textcolor{colorprop}{5} \times \textcolor{colordef}{3} \times \textcolor{colorprop}{5} \\ &= (\textcolor{colordef}{3} \times \textcolor{colordef}{3}) \times (\textcolor{colorprop}{5} \times \textcolor{colorprop}{5}) \\ &= \textcolor{colordef}{3}^{\textcolor{olive}{2}}\, \textcolor{colorprop}{5}^{\textcolor{olive}{2}}\end{aligned}$$In general, when you multiply two numbers \(\textcolor{colordef}{a}\) and \(\textcolor{colorprop}{b}\), and then raise the product to the power \(\textcolor{olive}{n}\), that is$$(\textcolor{colordef}{a}\textcolor{colorprop}{b})^{\textcolor{olive}{n}},$$the result is each factor raised to the power \(\textcolor{olive}{n}\):$$(\textcolor{colordef}{a}\textcolor{colorprop}{b})^{\textcolor{olive}{n}} = \textcolor{colordef}{a}^{\textcolor{olive}{n}}\, \textcolor{colorprop}{b}^{\textcolor{olive}{n}}.$$

Proposition — Exponent Law 4
For any numbers \(n\) and any numbers \(a\) and \(b\),$$\left(\textcolor{colordef}{a}\textcolor{colorprop}{b}\right)^{\textcolor{olive}{n}} = \textcolor{colordef}{a}^{\textcolor{olive}{n}}\, \textcolor{colorprop}{b}^{\textcolor{olive}{n}}$$
Example
Simplify \((\textcolor{colordef}{2}\times \textcolor{colorprop}{5})^{\textcolor{olive}{3}}\).

$$(\textcolor{colordef}{2}\times \textcolor{colorprop}{5})^{\textcolor{olive}{3}}= \textcolor{colordef}{2}^{\textcolor{olive}{3}}\, \textcolor{colorprop}{5}^{\textcolor{olive}{3}}$$

Skills to practice
  • Simplifying Powers of Products
  • Simplifying Powers of Products
VIII Exponent Law 5

Let's look at an example:$$\begin{aligned}\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{\textcolor{olive}{2}}&= \left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right) \times \left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right) \\ &= \dfrac{\textcolor{colordef}{5} \times \textcolor{colordef}{5}}{\textcolor{colorprop}{3} \times \textcolor{colorprop}{3}} \\ &= \dfrac{\textcolor{colordef}{5}^{\textcolor{olive}{2}}}{\textcolor{colorprop}{3}^{\textcolor{olive}{2}}}\end{aligned}$$In general, when a quotient \(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\) is raised to a power \(\textcolor{olive}{n}\), that is$$\left(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\right)^{\textcolor{olive}{n}},$$the result is the numerator raised to that power divided by the denominator raised to that power:$$\left(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\right)^{\textcolor{olive}{n}}= \dfrac{\textcolor{colordef}{a}^{\textcolor{olive}{n}}}{\textcolor{colorprop}{b}^{\textcolor{olive}{n}}}.$$

Proposition — Exponent Law 5
For \(b\neq 0\) and any number \(n\),$$\left(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\right)^{\textcolor{olive}{n}}= \dfrac{\textcolor{colordef}{a}^{\textcolor{olive}{n}}}{\textcolor{colorprop}{b}^{\textcolor{olive}{n}}}$$
Example
Calculate \(\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{\textcolor{olive}{2}}\).

$$\begin{aligned}[t]\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{\textcolor{olive}{2}}&= \dfrac{\textcolor{colordef}{5}^{\textcolor{olive}{2}}}{\textcolor{colorprop}{3}^{\textcolor{olive}{2}}} \\ &= \dfrac{25}{9}\end{aligned}$$

Skills to practice
  • Simplifying Powers of Fractions
  • Simplifying Powers of Algebraic Fractions
IX Exponent Law 6

Let's look at an example with a negative exponent:$$\begin{aligned}\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{\textcolor{olive}{-2}}&= \dfrac{1}{\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{\textcolor{olive}{2}}} \\ &= \dfrac{1}{\dfrac{\textcolor{colordef}{5}^{\textcolor{olive}{2}}}{\textcolor{colorprop}{3}^{\textcolor{olive}{2}}}} \\ &= 1 \times \dfrac{\textcolor{colorprop}{3}^{\textcolor{olive}{2}}}{\textcolor{colordef}{5}^{\textcolor{olive}{2}}} \\ &= \dfrac{\textcolor{colorprop}{3}^{\textcolor{olive}{2}}}{\textcolor{colordef}{5}^{\textcolor{olive}{2}}} \\ &= \left(\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{5}}\right)^{\textcolor{olive}{2}}\end{aligned}$$In general, when a quotient \(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\) is raised to a negative power \(\textcolor{olive}{-n}\),$$\left(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\right)^{\textcolor{olive}{-n}} = \left(\dfrac{\textcolor{colorprop}{b}}{\textcolor{colordef}{a}}\right)^{\textcolor{olive}{n}}.$$This means that a negative exponent makes the fraction flip: the numerator and denominator swap places.

Proposition — Exponent Law 6
For non-zero numbers \(a\) and \(b\), and any number \(n\),$$\left(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\right)^{\textcolor{olive}{-n}} = \left(\dfrac{\textcolor{colorprop}{b}}{\textcolor{colordef}{a}}\right)^{\textcolor{olive}{n}}$$and in particular,$$\left(\dfrac{\textcolor{colordef}{a}}{\textcolor{colorprop}{b}}\right)^{\textcolor{olive}{-1}} = \dfrac{\textcolor{colorprop}{b}}{\textcolor{colordef}{a}}.$$
Example
Calculate \(\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{-2}\).

$$\begin{aligned}\left(\dfrac{\textcolor{colordef}{5}}{\textcolor{colorprop}{3}}\right)^{-2} &= \left(\dfrac{\textcolor{colorprop}{3}}{\textcolor{colordef}{5}}\right)^{2} \\ &= \dfrac{3^2}{5^2}\\ &= \dfrac{9}{25} \\ \end{aligned}$$

Skills to practice
  • Expressing Negative Exponents as Fractions
  • Multiplying by the Inverse
X Order of operations
The order of operations is a set of rules that tells us which calculations to do first in a mathematical expression.
Definition — Order of Operations
To solve mathematical expressions accurately, we follow the order of operations, which is commonly remembered using the acronym PEMDAS:
  1. P: Parentheses
  2. E: Exponents
  3. M: Multiplication
  4. D: Division
  5. A: Addition
  6. S: Subtraction
We first do the operations at the top of the list.Multiplication and division are on the same level, so we work from left to right.Addition and subtraction are also on the same level, so we again work from left to right.
Example
Evaluate \((1+2) \times 2^3 + 4\).

$$\begin{aligned}[t](1+2) \times 2^3 + 4 &= \textcolor{colordef}{(1+2)} \times 2^3 + 4 && (\text{parentheses: } \textcolor{colordef}{(1+2)} = 3) \\ &= 3 \times \textcolor{colordef}{2^3} + 4 && (\text{exponent: } \textcolor{colordef}{2^3} = 8) \\ &= \textcolor{colordef}{3 \times 8} + 4 && (\text{multiplication: } \textcolor{colordef}{3 \times 8} = 24) \\ &= \textcolor{colordef}{24 + 4} && (\text{addition: } \textcolor{colordef}{24 + 4} = 28) \\ &= 28\end{aligned}$$

Skills to practice
  • Evaluating Expressions with Exponents in 2 Steps
  • Evaluating Expressions with Exponents in 3 Steps
  • Finding the Operators
  • Combining Negative Powers with Arithmetic
  • Simplifying Algebraic Expressions
  • Simplifying Expressions of Powers
  • Evaluating to an Integer
XI Scientific Notation
Working with very large or very small numbers can be awkward. Since our number system is base ten, we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. This way of writing numbers is called scientific notation and is especially useful in science.
Definition — Scientific Notation
A non-zero number is expressed in scientific notation when it is written in the form:
\(a \times 10^n\) where \(1 \leq |a| < 10\) and \(n\) is an integer.
Example
Write \(245\) in scientific notation.

$$\begin{aligned}[t]245 &= 2.45 \times 100 \\ &= 2.45 \times 10^2\end{aligned}$$So \(245\) in scientific notation is \(2.45 \times 10^2\).

Skills to practice
  • Writing Numbers as Powers of Ten
  • Expressing Numbers in Scientific Notation
  • Expressing in Decimal Form
  • Expressing Real-World Quantities in Scientific Notation
XII Exponential Expression
Definition — Exponential Expression
An exponential expression is a mathematical expression where a variable appears in the exponent.
Example
\(2^x\) and \(5^{x+1}\) are exponential expressions. This is different from a polynomial expression like \(x^2\), where the variable is in the base.
Method — Manipulating Exponential Expressions
Applying the exponent laws allows us to simplify, expand, and factorize complex expressions involving variables in the exponent. These skills are fundamental for solving exponential equations.
Example
Simplify \(\frac{2^{x+1}+2^x}{2^x}\).

$$\begin{aligned}[t] \frac{2^{x+1}+2^x}{2^x} &= \frac{2^x \cdot 2^1 + 2^x}{2^x} && \scriptscriptstyle\text{(Using exponent law)} \\ &= \frac{2^x(2+1)}{2^x} && \scriptscriptstyle\text{(Factor out the common term } 2^x\scriptscriptstyle\text{)} \\ &= 3 && \scriptscriptstyle\text{(Cancel the common factor)} \end{aligned}$$

Skills to practice
  • Simplifying using Exponent Laws
  • Simplifying Exponential Expressions
  • Expanding and Simplifying Exponential Expressions
  • Factorizing Exponential Expressions
XIII The Exponential Number \(e\)

Let’s explore an idea from finance: compound interest.
Imagine you invest \(\dollar 1\) in a special bank account that offers a \(100\pourcent\) annual interest rate. We will see how much money you have after one year, depending on how often the interest is calculated (compounded) and added to your account.
  • Case 1: Compounded annually
    The interest is paid once at the end of the year. The value is:$$1 \times (1 + 100\pourcent) = 1 \times (1+1) = \dollar 2.$$So$$\text{Value} = (1 + 1)^1.$$
  • Case 2: Compounded semi-annually
    The interest is paid twice a year. The bank gives you half the annual rate (\(50\pourcent\)) each time.
    • After 6 months: \(1 \times \left(1 + \dfrac{1}{2}\right) = \dollar 1.50\)
    • At the end of the year: \(1.50 \times \left(1 + \dfrac{1}{2}\right) = \dollar 2.25\)
    This can be calculated in one step as$$1 \times \left(1 + \frac{1}{2}\right)^2 = \dollar 2.25.$$So$$\text{Value} = \left(1 + \frac{1}{2}\right)^2.$$
  • Case \(n\): Compounded \(n\) times per year
    If we compound \(n\) times per year, the interest rate per period is \(\dfrac{1}{n}\) (since the total annual rate is \(100\pourcent\)), and it is applied \(n\) times. The value after one year is$$\text{Value} = \left(1 + \frac{1}{n}\right)^n.$$
Let’s calculate the value of \(\left(1 + \dfrac{1}{n}\right)^n\) for increasingly large values of \(n\):
Compounding Frequency \(n\) Value \(\left(1 + \dfrac{1}{n}\right)^n\)
Annually 1 \((1 + 1/1)^1 = 2.00000\)
Semi-annually 2 \((1 + 1/2)^2 = 2.25000\)
Quarterly 4 \((1 + 1/4)^4 \approx 2.44141\)
Monthly 12 \((1 + 1/12)^{12} \approx 2.61304\)
Daily 365 \((1 + 1/365)^{365} \approx 2.71457\)
Hourly 8,760 \((1 + 1/8760)^{8760} \approx 2.71813\)
As you can see, the final amount increases, but it does not grow without bound. It seems to approach a specific number. This special irrational number is denoted by \(e\), also known as Euler’s number. It represents the limiting value of this process of increasingly frequent compounding.

Definition — The Exponential Number \(e\)
The number e is defined as$$e = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n.$$
Correct to five decimal places,$$e \approx 2.71828\ldots$$It is an irrational number, which means its decimal representation never terminates and never repeats.
Skills to practice
  • Simplifying using Exponent Laws
  • Simplifying using Exponent Laws
  • Expanding and Simplifying Exponential Expressions
  • Expanding and Simplifying Exponential Expressions
  • Factorizing Exponential Expressions
  • Simplifying Exponential Expressions
XIV Exponential Equations
Definition — Exponential Equation
An exponential equation is an equation in which the unknown occurs as part of the index or exponent.
Example
\(2^x = 8\) and \(30 \times 3^x = 7\) are both exponential equations.
There are a number of methods we can use to solve exponential equations. These include graphing, using technology, and by using logarithms, which we will study later. However, in some cases we can solve the equation algebraically by equating indices.
Method — Solving by Equating Indices
For \(a>0, a\neq 1\), \(a^x = a^y\) if and only if \(x = y\).
Example
Solve for \(x\):
\(2^x = 16\)

$$\begin{aligned}[t]\,&& 2^x &= 16 \\ \Leftrightarrow && 2^x &= 2^4 && \scriptscriptstyle\text{(Write 16 as a power of 2)} \\ \Leftrightarrow && x &= 4 && \scriptscriptstyle\text{(Equate the indices)}\end{aligned}$$

Skills to practice
  • Solving by Equating Indices: Level 1
  • Solving by Equating Indices: Level 2
  • Solving by Equating Indices: Level 3
  • Solving Equations in Quadratic Form
XV Review \& Beyond
Skills to practice
  • Quiz
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