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 gives worked examples and practice-style 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 you evaluate an expression. The common acronym PEMDAS helps you remember the order:

1. Parentheses: evaluate expressions inside parentheses first.
2. Exponents: calculate exponents (powers and roots).
3. Multiplication and division: perform from left to right.
4. Addition and subtraction: perform from left to right.

Common mistakes

Many arithmetic errors come from misapplying the order of operations or misreading how an expression is grouped. These mistakes are common because they often seem intuitive, even though they break the formal rules. Knowing the patterns helps you slow down and evaluate expressions correctly.

Example: Ignoring parentheses

• Parentheses mark operations that must be completed first. Evaluate everything inside parentheses before applying operations outside them.
• Mistake: working outside the parentheses before simplifying inside.

Answer: $3\times (2+4)$ should be $3\times 6=18$, not $3\times 2+4=10$.

Example: Incorrect order of multiplication and division

• Multiplication and division have the same priority, so you work left to right as they appear.
• Mistake: always doing multiplication before division, regardless of order.

Answer: $8÷2\times 4$ should be $4\times 4=16$, not $8÷(2\times 4)=1$.

Example: Incorrect order of addition and subtraction

• Addition and subtraction also share the same priority, so you work left to right.
• Mistake: always doing addition before subtraction, regardless of order.

Answer: $10-3+2$ should be $7+2=9$, not $10-(3+2)=5$.

Example: Forgetting to apply the order of operations

• You must follow the full order of operations, not simple left-to-right evaluation. In particular, multiplication must be completed before addition.
• Mistake: evaluating strictly left to right.

Answer: $2+3\times 4$ should be $2+12=14$, not $(2+3)\times 4=20$.

Example: Misinterpreting exponents

• An exponent applies only to the value immediately before it unless parentheses show otherwise. A leading negative sign is not automatically included in the exponent.
• Mistake: treating $-2^2$ as if the base were $(-2)$.

Answer: $-2^2$ should be $-(2^2)=-4$, not $(-2{)}^2=4$.

Common number properties

Property	Example	Description
Commutative	$a+b=b+a$; $ab=ba$	You can switch the order of addition or multiplication.
Associative	$(a+b)+c=a+(b+c)$	Grouping of addition or multiplication does not affect the result.
Distributive	$a(b+c)=ab+ac$	Multiply a number across terms inside parentheses.
Identity	$a+0=a$; $a\times 1=a$	Zero is additive identity; one is multiplicative identity.
Inverse	$a+(-a)=0$; $a\times \frac{1}{a}=1$ (for $a\ne 0$)	Using opposites or reciprocals to undo operations.

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.

Example: Using exponent rules

• $2^3\cdot 2^4=2^{3+4}=2^7$
• $\frac{5^6}{5^2}=5^{6-2}=5^4$
• $(3^2{)}^3=3^{2\cdot 3}=3^6$
• $7^0=1$
• $4^{-2}=\frac{1}{4^2}=\frac{1}{16}$

Strategy: problem solving with variables

• Simplify expressions using PEMDAS.
• Look for opportunities to use number properties to simplify.
• Substitute values when needed (for example, test values for variables).
• Isolate variables when solving equations (undo operations in reverse).

Example: Simplifying with PEMDAS Simplify $8+2\times (3^2-5)$.

(spoiler)

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

Answer: $16$

Example: Apply the distributive property Simplify $5(x+2)-3x$.

(spoiler)

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

Answer: $2x+10$

Example: Solve using inverse operations Solve $\frac{3x-2}{4}=5$.

• Multiply both sides by $4$: $3x-2=20$
• Add $2$ to both sides: $3x=22$
• Divide by $3$: $x=\frac{22}{3}$

Answer: $x=\frac{22}{3}$

Example: Word problem involving properties If the sum of twice a number and $3$ is equal to $15$, what is the number? Let $x$ be the number.

(spoiler)

• $2x+3=15$
• Subtract $3$: $2x=12$
• Divide by $2$: $x=6$

Answer: $6$

Operations with square roots

Square roots represent the side length of a square with a given area. When a number is not a perfect square, its square root isn’t a whole number, but you can still estimate it and simplify it into an exact radical form. Estimation helps you check reasonableness; simplification helps you keep answers exact.

A square root like $\sqrt{10}$ isn’t a perfect square, but you can estimate it by locating it between nearby perfect squares.

Example: Estimating a square root Estimate the value of $\sqrt{10}$.

• $\sqrt{9}=3$ and $\sqrt{16}=4$, so $\sqrt{10}$ lies between $3$ and $4$.

• Square nearby decimals:

  • $3.1^2=9.61$
  • $3.2^2=10.24$

• So $3.1<\sqrt{10}<3.2$. Since $10$ is closer to $10.24$ than to $9.61$, the value should be closer to $3.2$ than to $3.1$.

• A calculator gives an approximation close to $3.16$.

Answer: $\sqrt{10}\approx 3.16$

Exact answers are usually written in simplified radical form. To simplify a square root, factor the radicand and pull out any perfect-square factors.

Example: Simplify a square root $\sqrt{18}=\sqrt{9\cdot 2}=\sqrt{9}\sqrt{2}=3\sqrt{2}$

Answer: $3\sqrt{2}$

Sidenote

How to simplify square roots

When simplifying square roots, always look for a factor that is a perfect square.

For example, $18=3\cdot 6$ does not help, because neither factor is a perfect square.

Using $18=9\cdot 2$ allows the square root to simplify.

Example: Simplify a square root Simplify $\sqrt{50}$.

Simplifying expressions with square roots

When square roots appear in expressions, combining them works like combining like terms in algebra: you can add or subtract radicals only when they have the same radicand. Before you try to combine anything, simplify each radical as much as possible.

Example: Combine like radicals $6\sqrt{3}-2\sqrt{3}=4\sqrt{3}$

Answer: $4\sqrt{3}$

Example: Unlike radicals do not combine $2\sqrt{5}+3\sqrt{7}$ cannot be combined because the radicands are different.

Answer: $2\sqrt{5}+3\sqrt{7}$

Sometimes radicals that look different become like radicals after simplification.

Example: Simplify before combining $\sqrt{12}-\sqrt{3}$

• $\sqrt{12}=\sqrt{4\cdot 3}=2\sqrt{3}$
• $2\sqrt{3}-\sqrt{3}=\sqrt{3}$

Answer: $\sqrt{3}$

Sidenote

Always simplify square roots before adding or subtracting

Square roots can only be added or subtracted when they have the same radicand. Always simplify each radical first before combining terms, since simplification may reveal like radicals that were not obvious at first.

For example, $\sqrt{12}-\sqrt{3}$ cannot be combined as written, but simplifying gives $2\sqrt{3}-\sqrt{3}$, which can be combined.

In contrast, $2\sqrt{5}+3\sqrt{7}$ cannot be combined even after simplification because the radicands remain different.

Multiplying square roots

When multiplying square roots, you can multiply the numbers inside the radicals using the product property of square roots. After multiplying, simplify the result if possible.

If $a\ge 0$ and $b\ge 0$, then

$$\sqrt{a}\sqrt{b}=\sqrt{ab}.$$

Example: Multiply radicals $\sqrt{2}\cdot \sqrt{3}=\sqrt{6}$

Answer: $\sqrt{6}$

When coefficients are present, multiply the coefficients separately from the radicals.

Example: Multiply radicals and simplify $2\sqrt{3}\cdot 4\sqrt{6}$

• Multiply coefficients: $2\cdot 4=8$
• Multiply radicals: $\sqrt{3}\sqrt{6}=\sqrt{18}=3\sqrt{2}$
• Combine: $8\cdot 3\sqrt{2}=24\sqrt{2}$

Answer: $24\sqrt{2}$

Example: Multiply radicals Multiply $3\sqrt{3}\times 4\sqrt{8}$.

Rationalizing denominators

Expressions are typically written so that no square root appears in the denominator. Rationalizing the denominator means rewriting the expression so the denominator is a rational number.

If the denominator contains a single square root, multiply the numerator and denominator by that root.

Example: Rationalize a denominator $\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

Answer: $\frac{\sqrt{2}}{2}$

Example: Rationalize a denominator Rationalize $\frac{5}{\sqrt{6}}$.

When the denominator contains a sum or difference involving a square root, multiply by the conjugate.

Example: Rationalize using a conjugate $\frac{1}{3+\sqrt{2}}\cdot \frac{3-\sqrt{2}}{3-\sqrt{2}}$

• $=\frac{3-\sqrt{2}}{(3+\sqrt{2})(3-\sqrt{2})}$
• $=\frac{3-\sqrt{2}}{9-2}$
• $=\frac{3-\sqrt{2}}{7}$

Answer: $\frac{3-\sqrt{2}}{7}$

Conjugates work because they create a difference of squares:

$$(a+b)(a-b)=a^2-b^2.$$

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 $B$.

Example: Union of sets Find $A\cup B$ where $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$.

Answer: $A\cup B=\{1,2,3,4,5,6\}$

Example: Intersection of sets Find $A\cap B$ where $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$.

Answer: $A\cap B=\{3,4\}$

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

(spoiler)

• 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.

Consider two sets, $A$ and $B$:

• $A\cap B$: elements in both sets, with count $n_{AB}$.
• $A\cup B$: elements in at least one set, with count $n(A\cup B)=n_A+n_B-n_{AB}$.
• $A$ only: count $n_A-n_{AB}$.
• $B$ only: count $n_B-n_{AB}$.
• Neither: count $N-n_A-n_B+n_{AB}$, where $N$ is the total number of elements in the universal set.

Solving problems with Venn diagrams

To solve problems using Venn diagrams, follow these steps:

1. Identify the sets: determine the sets involved and what they represent.
2. Draw the diagram: draw circles for each set, making sure to overlap them appropriately.
3. Label the regions: label each region with the appropriate counts or descriptions.
4. Fill in the counts: use the given information to fill in the counts for each region.
5. Solve the problem: use the diagram to answer the question or solve the problem.

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?

• Both: $20$
• Only math: $60-20=40$
• Only science: $40-20=20$
• Neither: $100-60-40+20=20$

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

Example: Book club attendance Out of $50$ members, $30$ read fiction, $22$ read nonfiction, and $12$ read both. How many are in each region?

(spoiler)

• Fiction only: $30-12=18$
• Nonfiction only: $22-12=10$
• Both: $12$
• Neither: $50-(18+10+12)=10$

Answer: fiction only $18$, nonfiction only $10$, both $12$, neither $10$

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

(spoiler)

• Rides only: $100-40=60$
• Games only: $75-40=35$
• Exactly one: $60+35=95$

Answer: $95$