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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.

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NEW! S-219, Firing Operations (Blended) Now Available

Date: Sept 11, 2025
Questions? Please contact:
Fuels Management Committee 

The S-219, Firing Operations (Blended) course is now available on the Wildland Fire Learning Portal. Developed through the Incident Performance and Training Modernization (IPTM) effort, this training supports individuals working towards Firing Boss, Single Resource (FIRB) incident qualifications.

This blended course combines online and instructor-led training to provide students with the knowledge and skills required to perform the duties of the FIRB, as described in the NWCG Incident Position Standards for Firing Boss, Single Resource, PMS 350-105.

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-219, Firing Operations (Blended)

NWCG Firing Boss, Single Resource Position Page

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NWCG Committees Observe Suicide Prevention Awareness Week: September 7-13, 2025

Date: Sep 5, 2025
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The NWCG Mental Health Subcommittee (MHSC) and Risk Management Committee (RMC) recognize the unique challenges and demands faced by the wildland fire community. The MHSC and RMC know that we are all stronger together and believe in fostering a culture of support, understanding, and resilience.

Suicide Prevention Awareness encourages us to actively connect with and support each other. For more information that could make a difference for yourself or someone else please review the materials and resources provided for the 2025 Suicide Prevention Awareness Week.

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2025 Suicide Prevention Awareness Week Materials and Resources

NEW! Incident Performance and Training Modernization Story Map Now Available

Date: Aug 26, 2025
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NWCG is excited to announce the launch of the Incident Performance and Training Modernization (IPTM) Story Map, a new interactive resource that showcases how NWCG is transforming wildland fire training and qualifications through a performance-based system.

Since its launch in 2023, IPTM has made major strides in updating how we manage incident positions:

  • 36 incident positions now have new Incident Position Standards and Next Generation Position Task Books available for field use
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The Story Map guides you through the IPTM process, highlights key accomplishments, and outlines what’s ahead. Whether you are a trainer or a trainee, this tool offers valuable insight into the modernization effort shaping the future of wildland fire training and qualifications.

This effort would not be possible without the contributions of NWCG staff, NWCG committees, and hundreds of subject matter experts across the wildland fire community.

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IPTM Story Map

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

Date: Aug 15, 2025
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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).

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S-290, Intermediate Wildland Fire Behavior (Blended)

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