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This presentation provides a drill on addition or subtraction of monomials as a practice on the beginning of the slides. It also presents the definition of sequence, arithmetic and geometric sequence with their examples and an activity to perform.
The presentation has first a drill on signed numbers. Then, it provides a definition examples and activities for the topics, " Finding the nth term of an Arithmetic Sequence, Arithmetic Mean and Arithmetic Series.".
This presentation provides a drill on addition or subtraction of monomials as a practice on the beginning of the slides. It also presents the definition of sequence, arithmetic and geometric sequence with their examples and an activity to perform.
The presentation has first a drill on signed numbers. Then, it provides a definition examples and activities for the topics, " Finding the nth term of an Arithmetic Sequence, Arithmetic Mean and Arithmetic Series.".
Fjfjdjejrkdidjxjsjrkkdjdkkjjkkfkckckdkükememememdjdjdjjdnrnene3irjrjrjüjejsjdndnenejejdjdjdjdjdjsjsjejrururirjrjrjddjdjjdjjrjrjrjrjriejrjrjridududjrjrjejejeididjdjdje**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit
You will learn how to get the value of a, b and c given a quadratic equations.
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Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
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In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
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Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
3. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Exponentes y
Productos
Notables
En esta semana repasamos los
temas de leyes de exponentes y
productos notables, necesarios
para los siguientes temas del
curso.
𝑎𝑚
. 𝑎𝑛
= 𝑎𝑚+𝑛
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
4. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
LEYES DE EXPONENTES
EXPONENTE NATURAL 𝑎𝑛
= 𝑎. 𝑎. 𝑎 … 𝑎
𝑛 𝑣𝑒𝑐𝑒𝑠
; 𝑛 ∈ ℕ 54
= 5.5.5.5 = 625
EXPONENTE CERO 𝑎0 = 1; 𝑎 ≠ 0 −
2
3
0
= 1
EXPONENTE NEGATIVO 𝑎−𝑛 =
1
𝑎𝑛
2
5
−3
=
5
2
3
=
53
23 =
125
8
EXPONENTE FRACCIONARIO 𝑎
𝑚
𝑛 =
𝑛
𝑎𝑚 8
2
3 =
3
82 =
3
8
2
= 22
5. C R E E M O S E N L A E X I G E N C I A
TEOREMAS
C U R S O D E Á L G E B R A
𝒂𝒎
. 𝒂𝒏
= 𝒂𝒎+𝒏
𝒂𝒎 𝒏
= 𝒂𝒎.𝒏
𝒂. 𝒃 𝒏
= 𝒂𝒏
. 𝒃𝒏
𝒂
𝒃
𝒏
=
𝒂𝒏
𝒃𝒏
𝒂𝒎
𝒂𝒏 = 𝒂𝒎−𝒏
𝒎 𝒏
𝒂 = 𝒎.𝒏
𝒂
𝒏
𝒂. 𝒃 = 𝒏
𝒂.
𝒏
𝒃
𝒏 𝒂
𝒃
=
𝒏
𝒂
𝒏
𝒃
𝑎7. 𝑎5 = 𝑎7+5 = 𝑎12 𝑎8
𝑎3
= 𝑎8−3 = 𝑎5
𝑎7 4 = 𝑎7.4 = 𝑎28
2𝑥 5
= 25
. 𝑥5
2
3
5
=
25
35
=
32
243
3 4
𝑎 = 3.4
𝑎 = 12
𝑎
3
8𝑥 =
3
8. 3
𝑥 = 23
𝑥
3 8
125
=
3
8
3
125
=
2
5
Ejemplos: Ejemplos:
TEOREMAS
6. C R E E M O S E N L A E X I G E N C I A
TEOREMAS
C U R S O D E Á L G E B R A
5
𝑥3 4
𝑥5
𝒂
𝒙𝒎𝒃
𝒙𝒏 =
𝒂𝒃
𝒙𝒎𝒃+𝒏
Ejemplo:
RADICALES INFINITOS
𝑴 =
𝒏
𝒂
𝒏
𝒂 𝒏
𝒂 …
𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒐𝒔 𝒓𝒂𝒅𝒊𝒄𝒂𝒍𝒆𝒔
Ejemplo: Calcule el valor de T
𝑇 =
3
25
3
25
3
25 …
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑠 𝑟𝑎𝑑𝑖𝑐𝑎𝑙𝑒𝑠
=
3
25. 𝑇
𝑇3
𝑇
= 25 𝑇 = 5
=
5.4
𝑥3.4+5 =
20
𝑥17
→ 𝑴 =
𝒏
𝒂. 𝑴 → 𝑇3
= 25𝑇
→ 𝑇2
= 25
7. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
RADICALES INFINITOS
𝑷 =
𝒏
𝒂 +
𝒏
𝒂 + 𝒏
𝒂 + ⋯
𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒐𝒔 𝒓𝒂𝒅𝒊𝒄𝒂𝒍𝒆𝒔
Ejemplo:
Calcule el valor de J
𝐽 = 12 + 12 + 12 + ⋯
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑠 𝑟𝑎𝑑𝑖𝑐𝑎𝑙𝑒𝑠
Solución:
𝐽 = 12 + 12 + 12 + ⋯
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑠 𝑟𝑎𝑑𝑖𝑐𝑎𝑙𝑒𝑠
= 12 + 𝐽
𝐽2
= 12 + 𝐽
𝐽 − 4 𝐽 + 3 = 0
Como 𝐽 > 0
→ 𝑷 =
𝒏
𝒂 + 𝑷
→ 𝐽 = 12 + 𝐽
→ 𝐽2
− 𝐽 − 12 = 0
→ 𝐽 = 4 ∨ 𝐽 = −3
→ 𝐽 = 4
8. C R E E M O S E N L A E X I G E N C I A
PRODUCTOS NOTABLES
C U R S O D E Á L G E B R A
PRODUCTO DE BINOMIOS CON TÉRMINO COMÚN
𝒙 + 𝒂 𝒙 + 𝒃 = 𝑥 + 5 𝑥 + 8 =
BINOMIO AL CUADRADO
𝒂 + 𝒃 𝟐 =
𝒂 − 𝒃 𝟐
=
3𝑥 + 5𝑦 2
=
= 9𝑥2
+ 30𝑥𝑦 + 25𝑦2
6 − 2
2
=
= 6 − 4 6 + 4
Ejemplos:
+ 𝒂 + 𝒃 𝒙
𝒙𝟐
+𝒂𝒃 𝑥2
+ 5 + 8 𝑥 +5.8 = 𝑥2 + 13𝑥 + 40
𝒂𝟐
+ 𝟐𝒂𝒃 + 𝒃𝟐
𝒂𝟐
− 𝟐𝒂𝒃 + 𝒃𝟐
3𝑥 2 + 2 5𝑦 2
3𝑥 5𝑦 +
6
2
−2 6 2 + 2 2
= 10 − 4 6
9. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
IDENTIDADES DE LEGENDRE
𝒂 + 𝒃 𝟐
+ 𝒂 − 𝒃 𝟐
=
𝒂 + 𝒃 𝟐 − 𝒂 − 𝒃 𝟐 =
6 + 5
2
+ 6 − 5
2
= 2 6 + 25
7 + 3
2
− 7 − 3
2
=
DIFERENCIA DE CUADRADOS
𝒂 + 𝒃 𝒂 − 𝒃 = 3𝑥 + 2𝑦 3𝑥 − 2𝑦
Ejemplos:
𝟐 𝒂𝟐
+ 𝒃𝟐
𝟒𝒂𝒃
= 2 6
2
+ 52
= 2 31 = 62
4 7 3 = 4 21
𝒂𝟐 − 𝒃𝟐 = 3𝑥 2
= 9𝑥2 − 4𝑦2
− 2𝑦 2
10. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
TRINOMIO AL CUADRADO
𝒂 + 𝒃 + 𝒄 𝟐
𝑎𝑏 + 𝑏𝑐 + 𝑎𝑐 2
BINOMIO AL CUBO
𝒂 + 𝒃 𝟑 𝑥 + 2 3
𝒂 − 𝒃 𝟑
𝑥 − 1 3
IDENTIDADES DE CAUCHY
𝒂 + 𝒃 𝟑 =
𝒂 − 𝒃 𝟑 =
𝑥 + 2 3
=
𝑥 − 1 3 =
Ejemplos:
Ejemplos:
Ejemplos:
= 𝒂𝟐
+ 𝒃𝟐
+ 𝒄𝟐 +𝟐 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂 = 𝑎𝑏 2
+ 𝑏𝑐 2
+ 𝑎𝑐 2
+2𝑎𝑏𝑐 𝑎 + 𝑏 + 𝑐
= 𝒂𝟑 + 𝟑𝒂𝟐𝒃 + 𝟑𝒂𝒃𝟐 + 𝒃𝟑
= 𝒂𝟑 − 𝟑𝒂𝟐𝒃 + 𝟑𝒂𝒃𝟐 − 𝒃𝟑
= 𝑥3
+ 3. 𝑥2
. 2 + 3. 𝑥. 22
+ 23
= 𝑥3 + 6𝑥2 + 12𝑥 + 8
= 𝑥3
− 3. 𝑥2
. 1 + 3. 𝑥. 12
− 13
= 𝑥3
− 3𝑥2
+ 3𝑥 − 1
𝒂𝟑
+ 𝒃𝟑
+ 𝟑𝒂𝒃 𝒂 + 𝒃
𝒂𝟑
− 𝒃𝟑
− 𝟑𝒂𝒃 𝒂 − 𝒃
𝑥3
+ 23
+ 3. 𝑥. 2. (𝑥 + 2)
𝑥3 − 13 − 3. 𝑥. 1. 𝑥 − 1
11. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
SUMA Y DIFERENCIA DE CUBOS
𝑎3
+ 𝑏3
=
𝑎3 − 𝑏3 =
𝑥 − 1 𝑥2 + 𝑥 + 1 =
𝑥 + 1 𝑥2
− 𝑥 + 1 =
IGUALDADES CONDICIONALES
Si: 𝑎 + 𝑏 + 𝑐 = 0
𝑎2 + 𝑏2 + 𝑐2 = −2 𝑎𝑏 + 𝑏𝑐 + 𝑎𝑐
𝑎3 + 𝑏3 + 𝑐3 = 3𝑎𝑏𝑐
Ejemplos:
Si 𝑎 = 4 − 7; 𝑏 = 7 − 1; 𝑐 = −3
Calcule el valor de
𝑀 =
𝑎3 + 𝑏3 + 𝑐3
𝑎𝑏
Resolución:
𝑎 + 𝑏 + 𝑐 = 0
Entonces, lo que nos piden queda:
𝑎3
+ 𝑏3
+ 𝑐3
𝑎𝑏
𝑎 + 𝑏
𝑎 − 𝑏
𝑎2
− 𝑎𝑏 + 𝑏2
𝑎2 + 𝑎𝑏 + 𝑏2
𝑥3
− 1
𝑥3 + 1
De los datos, se observa que
→ 𝑎3 + 𝑏3 + 𝑐3 = 3𝑎𝑏𝑐
=
3𝑎𝑏𝑐
𝑎𝑏
= 3𝑐 = 3 −3 = −9
12. C R E E M O S E N L A E X I G E N C I A
IGUALDADES CONDICIONALES
C U R S O D E Á L G E B R A
Si a, b, c son números reales, tales que:
𝒂𝟐
+ 𝒃𝟐
+ 𝒄𝟐
= 𝟎
Ejemplo
Si 𝑎, 𝑏 , 𝑐 son números reales y además
𝑎2
+ 𝑏2
+ 𝑐2
= 2𝑎 + 4𝑏 + 6𝑐 − 14
Calcule el valor de
𝑎 + 𝑏 + 𝑐.
Resolución
Todo al primer miembro, tenemos:
𝑎2 + 𝑏2 + 𝑐2 − 2𝑎 − 4𝑏 − 6𝑐 + 14 = 0
𝑎2
− 2𝑎 + 𝑏2
− 4𝑏 + 𝑐2
− 6𝑐 + 14 = 0
𝑎2
− 2𝑎
𝑎 − 1 2
𝑎 − 1 = 0 ∧ 𝑏 − 2 = 0 ∧ 𝑐 − 3 = 0
𝑎 = 1 ∧ 𝑏 = 2 ∧ 𝑐 = 3
∴ 𝑎 + 𝑏 + 𝑐 = 1 + 2 + 3 = 6
↔ 𝒂 = 𝒃 = 𝒄 = 𝟎
+12
+𝑏2
− 4𝑏+22 +𝑐2 − 6𝑐 +32 +14 −12
−22 −32 = 0
+ 𝑏 − 2 2
+ 𝑐 − 3 2 = 0
13. C R E E M O S E N L A E X I G E N C I A
IGUALDADES CONDICIONALES
C U R S O D E Á L G E B R A
Si a, b, c son números reales, tales que:
𝒂𝟐
+ 𝒃𝟐
+ 𝒄𝟐
= 𝒂𝒃 + 𝒃𝒄 + 𝒄𝒂
Ejemplo
Si 𝑎, 𝑏, 𝑐 son números reales y además
𝑎2 + 𝑏2 + 4 − 𝑎𝑏 − 2𝑎 − 2𝑏 = 0
Calcule el valor de
𝑎2 + 𝑏2
Resolución
Tenemos:
𝑎2 + 𝑏2 + 4 − 𝑎𝑏 − 2𝑎 − 2𝑏 = 0
𝑎2
+ 𝑏2
+ 4 = 𝑎𝑏 + 2𝑎 + 2𝑏
𝑎2
+ 𝑏2
+ 22
= 𝑎𝑏 + 2𝑎 + 2𝑏
𝑎 = 𝑏 = 2
Luego:
𝑎2 + 𝑏2 = 22 + 22 = 4 + 4 = 8
∴
↔ 𝒂 = 𝒃 = 𝒄
14. w w w . a c a d e m i a c e s a r v a l l e j o . e d u . p e
