The order of operations, number properties, exponent rules, and square root rules come up constantly in algebra and real-world problem solving. This section reviews the key rules and common pitfalls, then works through examples to build accuracy and speed.

Order of operations

The order of operations is a set of rules that tells you what to do first when evaluating an expression. The acronym PEMDAS helps you remember the order:

1. Parentheses: evaluate expressions inside parentheses first.
2. Exponents and roots: calculate powers and roots.
3. Multiplication and division (equal priority): perform left to right - neither one comes before the other.
4. Addition and subtraction (equal priority): perform left to right - neither one comes before the other.

Common mistakes with PEMDAS: Always simplify inside parentheses before applying any outside operations - $3\times (2+4)=3\times 6=18$, not $3\times 2+4=10$. Also, a leading negative sign is not part of the base: $-2^2=-(2^2)=-4$, whereas $(-2{)}^2=4$.

Number properties

Common number properties

Property	Example	Description
Commutative	$a+b=b+a$; $ab=ba$	You can switch the order of addition or multiplication.
Associative	$(a+b)+c=a+(b+c)$	Regroups terms within the same operation - no new multiplication is introduced.
Distributive	$a(b+c)=ab+ac$	Multiplies a factor across a sum or difference; unlike associative, this always introduces a new multiplication.
Identity	$a+0=a$; $a\times 1=a$	Zero is the additive identity; one is the multiplicative identity.
Inverse	$a+(-a)=0$; $a\times \frac{1}{a}=1$ (for $a\ne 0$)	Use opposites or reciprocals to undo operations.

Exponent rules

Working with integer exponents

Rule	Example	Description
Product rule	$x^m\cdot x^n=x^{m+n}$	Add exponents when bases are the same.
Quotient rule	$\frac{x^m}{x^n}=x^{m-n}$	Subtract exponents when dividing like bases.
Power of a power rule	$(x^m{)}^n=x^{mn}$	Multiply exponents when raising a power to a power.
Zero exponent rule	$x^0=1$ (for $x\ne 0$)	Any nonzero number to the $0$ power is $1$.
Negative exponent rule	$x^{-n}=\frac{1}{x^n}$	Flip the base and make the exponent positive.

Scientific notation

Scientific notation expresses a number in the form $a\times 10^n$, where $1\le |a|<10$ and $n$ is an integer. This format is especially useful for very large or very small numbers.

• To convert a large number to scientific notation, move the decimal point to the left until one non-zero digit remains to the left. The number of places moved becomes a positive exponent.
• To convert a small number (between 0 and 1), move the decimal point to the right. The number of places moved becomes a negative exponent.

Example: Converting to scientific notation

Convert $4200$ and $0.0042$ to scientific notation.

• $4200$: move the decimal $3$ places left → $4.2\times 10^3$
• $0.0042$: move the decimal $3$ places right → $4.2\times 10^{-3}$

Answer: $4200=4.2\times 10^3$; $0.0042=4.2\times 10^{-3}$

Problem-solving with variables

Example: Simplifying with PEMDAS and distribution

Simplify (a) $8+2\times (3^2-5)$, then (b) $5(x+2)-3x$.

(a) PEMDAS:

• Parentheses: $3^2-5=9-5=4$
• Multiply: $2\times 4=8$
• Add: $8+8=16$

(b) Distributive property:

• Distribute: $5x+10-3x$
• Combine like terms: $(5x-3x)+10=2x+10$

Answer: (a) $16$; (b) $2x+10$

Fraction operations

To add or subtract fractions, you need a common denominator. To divide by a fraction, multiply by its reciprocal.

Example: Fraction operations

Add $\frac{1}{3}+\frac{1}{4}$, then divide $\frac{2}{3}÷\frac{4}{5}$.

Addition:

• Find a common denominator: the LCD of $3$ and $4$ is $12$.
• Rewrite each fraction: $\frac{1}{3}=\frac{4}{12}$ and $\frac{1}{4}=\frac{3}{12}$
• Add: $\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$

Division:

• Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{2}{3}\times \frac{5}{4}$
• You can cancel common factors before multiplying: the $2$ in the numerator and $4$ in the denominator share a factor of $2$, giving $\frac{1}{3}\times \frac{5}{2}=\frac{5}{6}$.

Answer: $\frac{1}{3}+\frac{1}{4}=\frac{7}{12}$; $\frac{2}{3}÷\frac{4}{5}=\frac{5}{6}$

Operations with square roots

Square roots fall under the Exponents and roots step of PEMDAS - evaluate them after parentheses, but before multiplication, division, addition, and subtraction. To simplify a square root, factor the radicand and pull out any perfect-square factors. You can add or subtract radicals only when they share the same radicand; simplify each radical first, since doing so may reveal like radicals.

Example: Square roots - evaluate, simplify, and combine

(a) Evaluate $3+\sqrt{16-7}$.   (b) Simplify $\sqrt{12}-\sqrt{3}$.

(a)

• Inside the radical: $16-7=9$
• Exponents/roots: $\sqrt{9}=3$
• Add: $3+3=6$

(b)

• Simplify $\sqrt{12}=\sqrt{4\cdot 3}=2\sqrt{3}$
• Combine like radicals: $2\sqrt{3}-\sqrt{3}=\sqrt{3}$

Answer: (a) $6$; (b) $\sqrt{3}$

The same parentheses-first logic that governs arithmetic applies to set operations as well: grouping determines which operation you perform first.

Unions and intersections of sets

Definitions

Union ($A\cup B$)

The set containing all elements that are in $A$, or in $B$, or in both.

Intersection ($A\cap B$)

The set containing all elements that are in both $A$ and $B$.

Just as parentheses control the order of arithmetic operations, they also control the order of set operations. In an expression like $(A\cup B)\cap C$, you evaluate the union inside the parentheses first, then intersect the result with $C$.

Example: Combined set operations

Find $(A\cup B)\cap C$ where $A=\{1,2,3\}$, $B=\{3,4,5\}$, and $C=\{2,3,5,7\}$.

• First, $A\cup B=\{1,2,3,4,5\}$.
• Then, $(A\cup B)\cap C=\{2,3,5\}$.

Answer: $\{2,3,5\}$

Understanding Venn diagrams

A Venn diagram uses overlapping circles to represent sets. The overlap shows the intersection (elements in both sets), and the non-overlapping parts show elements that belong to only one set.

To count elements in each region of a two-set Venn diagram, use the inclusion-exclusion equation: $|A\cup B|=|A|+|B|-|A\cap B|$. Plug in what you know, solve for what you don’t, then fill in each region.

Solving problems with Venn diagrams

Example: Math and science

In a group of $100$ students, $60$ like math, $40$ like science, and $20$ like both. How many like only math, only science, and neither?

Let $M$ = students who like math ($n(M)=60$) and $S$ = students who like science ($n(S)=40$), with $n(M\cap S)=20$.

• Inclusion-exclusion: $n(M\cup S)=60+40-20=80$
• Both: $20$
• Only math: $60-20=40$
• Only science: $40-20=20$
• Neither: $100-80=20$

Answer: only math $40$, only science $20$, neither $20$

Example: Fair attractions

At a fair, $100$ people tried rides, $75$ tried games, and $40$ tried both. How many tried exactly one?

• $n(A\cup B)=100+75-40=135$
• Rides only: $100-40=60$; Games only: $75-40=35$; Exactly one: $60+35=95$

Answer: $95$