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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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As we approach the 2025 Week of Remembrance (WOR), June 30–July 6, we dedicate this time to reflect on the past incidents from 2015 and honor the fallen through learning. Since its inception in 2014, WOR has honored wildland firefighters who made the ultimate sacrifice while encouraging critical discussions that reinforce lessons learned.

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ETC Equipment Bulletin: 25-003 Flame-Resistant Shirt Survey

Date: June 12, 2025
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The U.S. Forest Service National Technology and Development Program (NTDP) is seeking input on the Forest Service-designed flame-resistant (FR) shirt, currently available through the National Interagency Support Cache system and FedMall.

NTDP is conducting a product review to improve future FR shirt designs. As part of this effort, a short survey is now open for federal employees who have worn any of three recent FR shirt models. Feedback will guide design improvements to better meet the needs of wildland firefighters.

To comply with the Paperwork Reduction Act, responses are limited to federal employees. The survey is open through September 12, 2025. 

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NEW! S-130, Firefighter Training (Blended) Now Available

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The S-130, Firefighter Training (Blended), equips new wildland firefighters with essential skills through a mix of online learning and hands-on training. The course features a required online component and an instructor-led exercise, reinforcing online concepts with practical performance-based training.

The course builds competencies in wildland fire operations, safety, and equipment use, applicable across various operational environments. Both the online component and instructor-led field exercise must be completed to receive a certificate of completion. Students are required to finish the prerequisite S-190, Introduction to Wildland Fire Behavior, before enrolling in S-130, Firefighter Training (Blended).

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