Skip to main content

Firefighter Math: 1.7 Powers and Roots

POWERS AND EXPONENTS

When a number is multiplied by itself several times, instead of writing each repetition of the multiplication, it is easier to use exponential notation. For example, 12 × 12 × 12 × 12 × 12 × 12 would be 12 multiplied to itself 6 times or 126. The number being multiplied is called the base and the number of times it is multiplied by itself is called the exponent. In the above case, the base is 12 and the exponent is 6. When a number has an exponent of 2, it is said to be squared. When the exponent is 3, the number is said to be cubed. When the exponent is 4, the number is expressed as the base to the fourth power, and so on.

Example 1 - Write 6 × 6 × 6 in exponential notation and in words. Solve.

6 × 6 × 6 = "six cubed" or 63. 63 = 216.

SQUARE ROOT


Powers can also be worked backwards to find the square root of the number. The number whose root you want to find will appear under a square root sign: √x. The square root of a number x is the number that would have to be squared in order to get x.

For example, the square root of 4 would be 2, since 22 = 4.

Example 2 - Find √49

√49 = 7 because 7 × 7 = 49
 

Perfect squares are numbers having a square root that is an integer. Not all numbers are perfect squares. In these situations, divide to see whether the number can be separated into two factors: the first factor is a number that is a perfect square while the rest stays as a square root.

Example 3  - Find √18
Divide into √(9x2). The square root of 9 is an integer (3), so it can be taken outside of the square root symbol. The 2 remains under the square root symbol, because it is not a perfect square. √18 = 3 × √ 2.

NWCG Latest Announcements

2025/2026 Leadership Campaign Quarter 2 Now Available!

Date: February 13, 2026
Questions? 
Please contact: Leadership Committee

NWCG is excited to announce that the Quarter Two materials for the 2025/2026 Wildland Fire Leadership Campaign are now available! This annual initiative invites all leadership and wildland fire management students to engage with essential skills and knowledge needed to lead effectively in dynamic environments.

This quarter focuses on Leadership Level 2: New Leader (Conveying Intent). Conveying intent is a crucial leadership competency in the high-risk, dynamic environment of wildland firefighting. Months 4 - 6 will concentrate on how successfully conveying intent enhances situational awareness, promotes shared understanding, and empowers firefighters to exercise initiative and adapt to changing conditions.

References:

2025/2026 NWCG Leadership Campaign - Leadership Levels

Wildland Fire Leadership Development Program

Leadership Committee

Opportunity to Serve the Wildland Fire Community as an IPTM SME!

Date: February 10, 2026
Questions? 
Please contact: NWCG SME Specialist

NWCG is seeking experienced wildland firefighters and support personnel to serve as Subject Matter Experts (SMEs) for the Incident Performance and Training Modernization (IPTM) initiative. The intent of IPTM is to transition NWCG’s training and qualification system to a performance-based model through updated position descriptions, Incident Position Standards, Next Generation Position Task Books (Next Gen PTBs), and updates to training where needed.

NWCG is currently recruiting SMEs for the following positions:

  • Aircraft Timekeeper (ATIM)
  • Compensation/Claims Unit Leader (COMP)
  • Cost Unit Leader (COST)
  • Crew Representative (CREP)
  • Deck Coordinator (DECK)
  • Fire Effects Monitor (FEMO)
  • Field Observer (FOBS)
  • Geographic Information System Specialist (GISS)
  • Ordering Manager (ORDM)
  • Receiving/Distribution Manager (RCDM)
  • Staging Area Manager (STAM)

Projects begin April/May 2026 and require 15–20 hours per month for 6–8 months, including a five-day workshop in Boise, Idaho this May. Apply by February 27, 2026, at the link below and review the attached document for details.

References:

NWCG SME Interest Sign-up

IPTM Subject Matter Expert Duties and Expectations Document

Incident Performance and Training Modernization (IPTM)

NEW! M-280, Resource Advisor, Fireline (Instructor-Led) Course Available Now

Date: January 29, 2026
Questions? 
Please contact: Resource Advisor Subcommittee

NWCG is pleased to announce the new M-280, Resource Advisor, Fireline (Instructor-Led) course is now available. This training supports individuals working towards Resource Advisor, Fireline (REAF) incident qualifications.

M-280, Resource Advisor, Fireline (Instructor-Led) training prepares students to serve as a Resource Advisor, Fireline (REAF) as defined in NWCG Incident Position Standards for Resource Advisor, Fireline, PMS 350-112. This course presents scenarios that reflect the diverse challenges a REAF may encounter during an incident. Through these scenarios, participants gain hands-on experience and an understanding of REAF responsibilities in a controlled learning environment.

References:

Resource Advisor, Fireline Position Page

NWCG Position Standards for Resource Advisor, Fireline, PMS 350-112

M-280, Resource Advisor, Fireline (Instructor-Led)

Wildland Fire Learning Portal

2026 Updates to the NWCG Standards for Wildland Fire Position Qualifications, PMS 310-1

Date: January 23, 2026
Questions? 
Please contact: Incident and Position Standards Committee

The Incident and Position Standards Committee has updated the NWCG Standards for Wildland Fire Position Qualifications, PMS 310-1.

These updates address changes to incident position names and requirements for qualification, training, and experience. Updates related to the implementation of Complex Incident Management (CIM) and Incident Performance Training Modernization (IPTM) are also included.

The NWCG Wildland Fire Position Qualification Flowchart, PMS 308 has also been updated to reflect the recent changes.

References:

NWCG Standards for Wildland Fire Position Qualifications, PMS 310-1

NWCG Wildland Fire Position Qualification Flowchart, PMS 308

IPSC Memo 26-01: January 2026 Updates to the NWCG Standards for Wildland Fire Positions Qualifications, PMS 310-1