Chapter 05 of 10

Numerical Expressions

How do conventions make complicated calculations unambiguous?

Numerical expressions contain numbers and operations but no equals sign. Exponents compress repeated multiplication, the order of operations makes evaluation consistent, and prime factorization reveals the building blocks of whole numbers.

01

The big ideas

Understand these before you worry about speed.

Exponents represent repeated multiplication

The base is the repeated factor and the exponent tells how many copies of that factor are multiplied. An exponent does not mean multiply the base by the exponent.

Example34=3⋅3⋅3⋅3=81\displaystyle 3^4=3\cdot3\cdot3\cdot3=81.

Follow the order of operations

Evaluate grouping symbols, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Example18−2(32)=18−2(9)=0\displaystyle 18-2(3^2)=18-2(9)=0.

Prime factorization shows a number's structure

A prime number has exactly two positive factors. Repeatedly divide a composite number by primes until every factor is prime.

Example60=22⋅3⋅5\displaystyle 60=2^2\cdot3\cdot5.

GCF and LCM answer different questions

The greatest common factor is the largest factor quantities share. The least common multiple is the smallest positive multiple they share.

ExampleFor 12\displaystyle 12 and 18\displaystyle 18, GCF⁡=6\displaystyle \operatorname{GCF}=6 and LCM⁡=36\displaystyle \operatorname{LCM}=36.
02

Worked example

Follow the reason for each step—not just the symbols.

Problem

Evaluate 48÷(22+2)+3(5−1)\displaystyle 48\div(2^2+2)+3(5-1).

  1. Evaluate the exponent and grouping: 22+2=6\displaystyle 2^2+2=6 and 5−1=4\displaystyle 5-1=4.
  2. The expression becomes 48÷6+3(4)\displaystyle 48\div6+3(4).
  3. Multiply and divide from left to right: 8+12\displaystyle 8+12.
  4. Add the remaining terms.
Answer20\displaystyle 20
03

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

1

Evaluate 53\displaystyle 5^3.

Check answer125\displaystyle 125
2

Evaluate 24÷3+2(7−4)\displaystyle 24\div3+2(7-4).

Check answer14\displaystyle 14
3

Write the prime factorization of 84\displaystyle 84.

Check answer22⋅3⋅7\displaystyle 2^2\cdot3\cdot7
4

Find the GCF and LCM of 8\displaystyle 8 and 12\displaystyle 12.

Check answerGCF⁡=4\displaystyle \operatorname{GCF}=4, LCM⁡=24\displaystyle \operatorname{LCM}=24
04

Practice and tools

Use the resource that matches what you need next.

Worksheet

Order of Operations

Practice grouping, exponents, and operation order.

Open resource Worksheet

Exponents

Evaluate powers and introductory exponent patterns.

Open resource Calculator

Prime Factorization Tool

Build a prime factorization with repeated division.

Open resource Calculator

GCF & LCM Calculator

Compare shared and required prime factors.

Open resource