Skip to main content

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

NEW! S-290, Intermediate Wildland Fire Behavior (Blended) Available Now

Date: Aug 15, 2025
Questions? Please contact:
Fire Behavior Subcommittee

NWCG is excited to announce that the S-290, Intermediate Wildland Fire Behavior (Blended) training is now available on the Wildland Fire Learning Portal. S-290 (Blended) training combines online training and instructor-led training components that support individuals working towards any Single Resource Boss or Fire Effects Monitor incident qualifications.

This second course in the series collectively serves to develop fire behavior prediction knowledge and skills, and provides more detailed information on fuels, weather, and topography that affect wildland fire behavior for safety purposes.

Students are required to finish the prerequisite S-190, Introduction to Wildland Fire Behavior, before enrolling in S-290, Intermediate Wildland Fire Behavior (Blended).

References:

S-290, Intermediate Wildland Fire Behavior (Blended)

S-190, Introduction to Wildland Fire Behavior

Wildland Fire Learning Portal

NEW! S-271, Helicopter Crewmember (Blended) Now Available

Date: Aug 14, 2025
Questions? Please contact:
Interagency Helicopter Operations Subcommittee

The S-271, Helicopter Crewmember (Blended) is now available. Developed through the Incident Performance and Training Modernization (IPTM) effort, this training supports individuals working towards Helicopter Crewmember (HECM) incident qualifications. 

S-271, Helicopter Crewmember (Blended) training combines online training and instructor-led training components. This course is designed to provide students with the knowledge and skills necessary to perform the duties of the HECM position, as described in the NWCG Incident Position Standards for Helicopter Crewmember, PMS 350-22.

Any changes to qualification pathways will take effect with the next update of the NWCG Standards for Wildland Fire Position Qualifications, PMS 310-1, scheduled for January 2026.

References:

S-271, Helicopter Crewmember (Blended)

NWCG Helicopter Crewmember (HECM) Position Page

Wildland Fire Learning Portal

Incident Position Standards and Next Generation Position Task Books Now Available for FBAN and LTAN

Date: Aug 13, 2025
Questions? Please contact:
Fire Behavior Subcommittee

NWCG is excited to announce that Incident Position Standards and Next Generation Position Task Books are now available for Fire Behavior Analyst (FBAN) and Long Term Fire Analyst (LTAN).

The Performance Support Packages for these positions were developed as part of the Incident Performance and Training Modernization (IPTM) effort. These resources support trainees, qualified personnel, and evaluators in their respective roles.

Any changes to qualification pathways will take effect with the next update of the NWCG Standards for Wildland Fire Position Qualifications, PMS 310-1, scheduled for January 2026.

References:

NWCG Fire Behavior Analyst Position Page

NWCG Long Term Fire Analyst Page

ETC/RMC Safety Bulletin: 25-001 New Guidance on Laundering Wildland Fire Clothing to Reduce Contamination

Date: Aug 8, 2025
Questions? Please contact:
Equipment Technology Committee
 Risk Management Committee

The Equipment Technology Committee (ETC) and the Risk Management Committee (RMC) have issued Safety Bulletin 25-001: Laundering to Decontaminate Wildland Fire Clothing. Recent research revealed that wildland fire flame-resistant pants and shirts can be contaminated with chemicals from combustion byproducts, including carcinogens, and that common laundering practices can effectively remove these harmful contaminants from wildland firefighter clothing more effectively than previously understood. It is recommended to decontaminate wildland fire clothing as frequently as possible. 

Frequently Asked Questions about this new information and how to implement recommendations can be found on the NWCG Alerts page. Read the complete ETC/RMC Safety Bulletin: 25-001 to learn more. 

References:

NWCG Alerts

ETC Safety Bulletin: 25-001

Frequently Asked Questions (FAQ) on Laundering to Decontaminate Wildland Fire Clothing

Equipment Technology Committee

Risk Management Committee