Working with numbers

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. It closes with two problem-solving skills: testing a claim with a counterexample, and picking out what a real-life problem needs.

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) $\sqrt{3}$

Testing a claim with a counterexample

A claim about every number - “always,” “never,” “every” - is disproved by one case where it fails. That case is a counterexample. Examples that agree with a claim never prove it, but a single counterexample disproves it. To find one, try the numbers that most often break a rule: $0$, $1$, negative numbers, and fractions between $0$ and $1$.

Example: finding a counterexample

A student claims that squaring an integer always gives a result greater than the integer. Which case shows the claim is false: $3^2=9$, $(-4{)}^2=16$, or $1^2=1$?

• $9>3$ and $16>-4$, so those cases agree with the claim.
• $1^2=1$, and $1$ is not greater than $1$.

Answer: $1^2=1$

Solving real-life problems

Real-life problems often give more information than you need. Before calculating, decide what the question asks for, pick out the numbers that affect that answer, and choose the operations that connect them.

Example: choosing the relevant numbers

A caterer charges $\$18$ per guest plus a $\$75$ setup fee. The party has $30$ guests, lasts $4$ hours, and uses $12$ tables. What does the catering cost?

• Guests: $30\times \$18=\$540$
• Add the setup fee: $\$540+\$75=\$615$
• The length of the party and the number of tables don’t change the cost.

Answer: $\$615$

The context also decides which way to round. When a partial item won’t do the job - buses for a field trip, cans of paint - round up, even if the decimal is small. When only complete items count - full boxes packed, tickets you can afford - round down: with $\$50$ and tickets at $\$12$, you can buy $4$ tickets, since $50÷12\approx 4.17$.

Example: rounding to fit the context

$150$ students and $10$ chaperones are going on a field trip, and each bus holds $48$ people. How many buses are needed?

• People: $150+10=160$
• Buses: $160÷48\approx 3.33$
• Three buses would leave $16$ people behind, so round up.

Answer: $4$ buses