6.2 math homework answers

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Answer Key 6.2

• [latex]2x^4y^{-2}\cdot 2^4x^4y^{12}[/latex] [latex]2^5x^8y^{10}\Rightarrow 32x^8y^{10}[/latex]
• [latex]2a^{-2}b^{-3}\cdot 2^4b^{16}[/latex] [latex]32a^{-2}b^{13}\Rightarrow \dfrac{32b^{13}}{a^2}[/latex]
• [latex]2^4x^8y^8\cdot x^{-4}[/latex] [latex]2^4x^4y^8\Rightarrow 16x^4y^8[/latex]
• [latex]2x^{-3--3}y^{2-3}3^{-2}[/latex] [latex]2\cancel{x^0}y^{-1}3^{-2}\Rightarrow \dfrac{2}{9y}[/latex]
• [latex]3y^3\div [6x^7y^{-2}][/latex] [latex]3x^{-7}y^{3--2}\div 6[/latex] [latex]3x^{-7}y^5\div 6 \Rightarrow \dfrac{y^5}{2x^7}[/latex]
• [latex]2y\div y^8[/latex] [latex]2y^{1-8}[/latex] [latex]2y^{-7}\Rightarrow \dfrac{2}{y^7}[/latex]
• [latex]a^{16}\div 2b\Rightarrow \dfrac{a^{16}}{2b}[/latex]
• [latex]2^4a^8b^{12}\div a^{-1}[/latex] [latex]16a^{8+1}b^{12}[/latex] [latex]16a^9b^{12}[/latex]
• [latex]{2}^{-2}{y}^{8}\div {x}^{2}\Rightarrow \dfrac{{y}^{8}}{{2}^{2}{x}^{2}}\Rightarrow \dfrac{{y}^{8}}{{4x}^{2}}[/latex]
• [latex]2^4m^4n^{4--2}[/latex] [latex]16m^4n^6[/latex]
• [latex]2x^{-3}\div x^{-4}y^3[/latex] [latex]2x^{-3--4}y^{-3}\Rightarrow 2xy^{-3} \Rightarrow \dfrac{2x}{y^3}[/latex]
• [latex][2u^{-2}v^3\cdot 2^{-1}u^{-1}v^{-4}]\div 2u^{-4}[/latex] [latex][u^{-3}v^{-1}]\div 2u^{-4}[/latex] [latex]2^{-1}u^{-3--4}v^{-1}[/latex] [latex]2^{-1}uv^{-1}\Rightarrow \dfrac{u}{2v}[/latex]
• [latex]2y \div 2^{-1}y^{-4}[/latex] [latex]2y\cdot 2y^4 \Rightarrow 4y^5[/latex]
• [latex]b^{-1}\div 2a^{-3}b^2[/latex] [latex]b^{-1}\cdot 2^{-1}a^3b^{-2}\Rightarrow \dfrac{a^3}{2b^3}[/latex]
• [latex]2x^2yz \div [2x^4y^4z^{-2}z^4y^8][/latex] [latex]2x^2yz \div [2x^4y^{12}z^2][/latex] [latex]2x^2yz \cdot z^{-1}x^{-4}y^{-12}z^{-2}[/latex] [latex]x^{-2}y^{-11}z^{-1}\Rightarrow \dfrac{1}{x^2y^{11}z}[/latex]
• [latex]2a^{-3}b^2c^2b^6\div a^9b^{-6}c^9[/latex] [latex]2a^{-3}b^8c^2\cdot a^{-9}b^6c^{-9}[/latex] [latex]2a^{-12}b^{14}c^{-7}\Rightarrow \dfrac{2b^{14}}{a^{12}c^7}[/latex]
• [latex]2m^2p^2q^5 \div [2^3m^3\cdot 4^3p^6][/latex] [latex]2m^2p^2q^5 \div [2^3m^3\cdot 2^6p^6][/latex] [latex]2m^2p^2q^5 \div 2^9m^3p^6[/latex] [latex]2m^2p^2q^5\cdot 2^{-9}m^{-3}p^{-6}[/latex] [latex]2^{-8}m^{-1}p^{-4}q^5\Rightarrow \dfrac{q^5}{2^8mp^4}\Rightarrow \dfrac{q^5}{256mp^4}[/latex]
• [latex]y^{-1}x^4z^{-2}\div x^2y^3z^2[/latex] [latex]y^{-1}x^4z^{-2}\cdot x^{-2}y^{-3}z^{-2}[/latex] [latex]x^2y^{-4}z^{-4}\Rightarrow \dfrac{x^2}{y^4z^4}[/latex]
• [latex]2mpn^{-3}\div 2n^2n^{-12}p^6[/latex] [latex]2mpn^{-3}\div 2n^{-10}p^6[/latex] [latex]2mpn^{-3}\cdot 2^{-1}n^{10}p^{-6}[/latex] [latex]mp^{-5}n^7\Rightarrow \dfrac{mn^7}{p^5}[/latex]

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6.2E: Exercises

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Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

• \((\frac{1}{3})^2\)
• \((0.2)^4\)
• \((\frac{2}{9})^2\)
• \((0.5)^3\)
• \(\frac{4}{81}\)
• \((\frac{2}{5})^3\)
• \((0.7)^2\)
• \((\frac{3}{4})^3\)
• \((0.4)^3\)
• \(\frac{27}{64}\)
• \((−6)^4\)
• \(−6^4\)
• \((−2)^6\)
• \(−2^6\)
• \(−(\frac{1}{4})^4\)
• \((−\frac{1}{4})^4\)
• \(−(\frac{2}{3})^2\)
• \((−\frac{2}{3})^2\)
• \(−\frac{4}{9}\)
• \(\frac{4}{9}\)
• \(−0.5^2\)
• \((−0.5)^2\)

Exercise 10

• \(−0.1^4\)
• \((−0.1)^4\)
• −0.0001

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

Exercise 11

\(d^3·d^6\)

Exercise 12

\(x^4·x^2\)

Exercise 13

\(n^{19}·n^{12}\)

Exercise 14

\(q^{27}·q^{15}\)

Exercise 15

• \(4^5·4^9\)
• \(8^9·8\)

Exercise 16

• \(3^{10}·3^6\)
• \(5·5^{4}\)

Exercise 17

• \(y·y^3\)
• \(z^{25}·z^8\)

Exercise 18

• \(w^5·w\)
• \(u^{41}·u^{53}\)

Exercise 19

\(w·w^2·w^3\)

Exercise 20

\(y·y^3·y^5\)

Exercise 21

\(a^4·a^3·a^9\)

Exercise 22

\(c^5·c^{11}·c^2\)

Exercise 23

\(m^x·m^3\)

Exercise 24

\(n^y·n^2\)

\(n^{y+2}\)

Exercise 25

\(y^a·y^b\)

Exercise 26

\(x^p·x^q\)

\(x^{p+q}\)

In the following exercises, simplify each expression using the Power Property for Exponents.

Exercise 27

• \((m^4)^2\)
• \( (10^3)^6\)

Exercise 28

• \((b^2)^7\)
• \((3^8)^2\)

Exercise 29

• \((y^3)^x\)
• \((5^x)^y\)

Exercise 30

• \((x^2)^y\)
• \((7^a)^b\)

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

Exercise 31

• \((3xy)^2\)

Exercise 32

• \((4ab)^2\)
• \(16a^{2}b^{2}\)

Exercise 33

• \((−4m)^3\)
• \((5ab)^3\)

Exercise 34

• \((−7n)^3\)
• \((3xyz)^4\)
• \(−343n^3\)
• \(81x^{4}y^{4}z^{4}\)

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

Exercise 35

• \((y^2)^4·(y^3)^2\)
• \((10a^{2}b)^3\)

Exercise 36

• \((w^4)^3·(w^5)^2\)
• \((2xy^4)^5\)
• \(32x^{5}y^{20}\)

Exercise 37

• \((−2r^{3}s^2)^4\)
• \((m^5)^3·(m^9)^4\)

Exercise 38

• \((−10q^{2}p^4)^3\)
• \((n^3)^{10}·(n^5)^2\)
• \(−1000q^{6}p^{12}\)

Exercise 39

• \((3x)^{2}(5x)\)
• \((5t^2)^{3}(3t)^{2}\)

Exercise 40

• \((2y)^{3}(6y)\)
• \((10k^4)^{3}(5k^6)^{2}\)
• \(25,000k^{24}\)

Exercise 41

• \((5a)^{2}(2a)^3\)
• \((12y^2)^{3}(23y)^2\)

Exercise 42

• \((4b)^{2}(3b)^{3}\)
• \((12j^2)^{5}(25j^3)^2\)
• \(1200j^{16}\)

Exercise 43

• \((25x^{2}y)^3\)
• \((89xy^4)^2\)

Exercise 44

• \((2r^2)^{3}(4r)^2\)
• \((3x^3)^{3}(x^5)^4\)
• \(128r^{8}\)
• \(27x^{29}\)

Exercise 45

• \((m^{2}n)^{2}(2mn^5)^4\)
• \((3pq^4)^{2}(6p^{6}q)^2\)

In the following exercises, multiply the monomials.

Exercise 46

\((6y^7)(−3y^4)\)

\(−18y^{11}\)

Exercise 47

\((−10x^5)(−3x^3)\)

Exercise 48

\((−8u^6)(−9u)\)

\(72u^{7}\)

Exercise 49

\((−6c^4)(−12c)\)

Exercise 50

\((\frac{1}{5}f^8)(20f^3)\)

\(4f^{11}\)

Exercise 51

\((\frac{1}{4}d^5)(36d^2)\)

Exercise 52

\((4a^{3}b)(9a^{2}b^6)\)

\(36a^{5}b^7\)

Exercise 53

\((6m^{4}n^3)(7mn^5)\)

Exercise 54

\((\dfrac{4}{7}rs^2)(14rs^3)\)

\(8r^{2}s^5\)

Exercise 55

\((\dfrac{5}{8}x^{3}y)(24x^{5}y)\)

Exercise 56

\((\frac{2}{3}x^{2}y)(\frac{3}{4}xy^2)\)

\(\frac{1}{2}x^{3}y^3\)

Exercise 57

\((\dfrac{3}{5}m^{3}n^2)(\dfrac{5}{9}m^{2}n^3)\)

​​​​​​ Mixed Practice

Exercise 58

\((x^2)^4·(x^3)^2\)

Exercise 59

\((y^4)^3·(y^5)^2\)

Exercise 60

\((a^2)^6·(a^3)^8\)

Exercise 61

\((b^7)^5·(b^2)^6\)

Exercise 62

\((2m^6)^3\)

\(8m^{18}\)

Exercise 63

\((3y^2)^4\)

Exercise 64

\((10x^{2}y)^3\)

\(1000x^{6}y^3\)

Exercise 65

\((2mn^4)^5\)

Exercise 66

\((−2a^{3}b^2)^4\)

\(16a^{12}b^8\)

Exercise 67

\((−10u^{2}v^4)^3\)

Exercise 68

\((\frac{2}{3}x^{2}y)^3\)

\(\frac{8}{27}x^{6}y^3\)

Exercise 69

\((\frac{7}{9}pq^4)^2\)

Exercise 70

\((8a^3)^{2}(2a)^4\)

\(1024a^{10}\)

Exercise 71

\((5r^2)^{3}(3r)^2\)

Exercise 72

\((10p^4)^{3}(5p^6)^2\)

\(25000p^{24}\)

Exercise 73

\((4x^3)^{3}(2x^5)^4\)

Exercise 74

\((\frac{1}{2}x^{2}y^3)^{4}(4x^{5}y^3)^2\)

\(x^{18}y^{18}\)

Exercise 75

\((\frac{1}{3}m^{3}n^2)^{4}(9m^{8}n^3)^2\)

Exercise 76

\((3m^{2}n)^{2}(2mn^5)^4\)

\(144m^{8}n^{22}\)

Exercise 77

\((2pq^4)^{3}(5p^{6}q)^2\)

​​​​​​ Everyday Math

Exercise 78

Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is \(10^2\), on the third round is \(10^3\), as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Exercise 79

Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $35,000(1.03), after 2 years was $\(35,000(1.03)^2\), after 3 years was $\(35,000(1.03)^3\), as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.

Exercise 80

Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $1,000(0.70) and the cost of the item during the second week would be $\(1,000(0.70)^2\). Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Exercise 81

Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for $20,000, the value at the end of the first year would be $20,000(0.90) and the value of the car after the end of the second year would be $\(20,000(0.90)^2\). Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Writing Exercises

Exercise 82

Use the Product Property for Exponents to explain why \(x·x=x^2\)

Answers will vary.

Exercise 83

Explain why \(−5^3=(−5)^3\), but \(−5^4 \ne (−5)^4\).

Exercise 84

Jorge thinks \((\frac{1}{2})^2\) is 1. What is wrong with his reasoning?

Exercise 85

Explain why \(x^3·x^5\) is \(x^8\), and not \(x^{15}\).

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. After reviewing this checklist, what will you do to become confident for all goals?

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Google's Circle to Search can help students with their homework, here is how to use the feature

I n January this year, Google announced the Circle To Search feature for Android devices, which allowed users to search for anything on their phones without having to switch applications. Users can simply outline items on their screen with their fingers to prompt a Google search. The feature debuted on Samsung's Galaxy S24 series and was then introduced on the Pixel 8, Pixel 8 Pro, and the recently-launched Pixel 8a. Last night, during Google's annual developer conference, Google I/O, the company introduced new updates for the Circle to Search feature. 

Google is boosting the capabilities of Android's Circle to Search and it can now prove to be useful for students. The feature now includes a new tool designed to guide students through solving math and physics problems from their homework. Google says that while the feature aids in understanding the problem, it doesn't complete the homework for students and is there to help them learn how to tackle the problems themselves.

By using an Android smartphone or tablet, students can activate Circle to Search to receive AI-driven assistance with mathematical word problems. The function breaks down the problem and outlines steps necessary to find the correct solution. 

This enhancement comes at a time when the use of AI tools in education, like ChatGPT, has sparked debate. There are widespread concerns about how students might use these tools to expedite their homework. Google is keen to position this new capability as a supportive educational tool, addressing some of these concerns by focusing on helping students learn rather than doing their work for them.

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Circle to Search is the new cheat sheet for your Math and Physics homework

Published on May 14, 2024

• Circle to Search can now help students with homework, making it easy to search for instructions and answers without leaving the digital homework screen.
• Later this year, Circle to Search will be able to solve even more complex problems involving symbolic formulas, diagrams, graphs, and more.

Circle to Search was one of the flagship features of the Galaxy S24 series . It lets users easily access Google Search on practically any screen. The feature then expanded to the Pixel 8 series and then to other Samsung Galaxy phones and Pixel devices. If you have Circle to Search on your phone, you can now use it to help with your homework.

As part of its Google I/O 2024 announcements, Google has announced that Circle to Search can now help students with homework. When students need help, they can open their camera or their digital homework on their phones or tablets and circle the question they are stuck on. Circle to Search will then present a step-by-step instruction to solve their homework question.

Subjects covered include physics and match word problems. This will make searching far easier, as you no longer have to leave your digital homework screen or input the question yourself. The fine print says that right now, this is available for some math word problems when opted into Search Labs.

Google says that later this year, Circle to Search will be able to solve even more complex problems involving symbolic formulas, diagrams, graphs, and more. This will be made possible through LearnLM, Google’s new family of AI models fine-tuned for learning.

Circle to Search is currently available on over 100 million devices, and Google aims to double its reach to 200 million devices by the end of 2024. With new features like homework help, Circle to Search becomes even more useful, and it is available only on (select) Android devices.

Do you use Google or other search engines to help with your homework? Will you now switch to Circle to Search for your homework? Let us know in the comments below!

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• She announced GPT-4o, its next flagship AI model, with improved voice and vision capabilities.

OpenAI just took the wraps off a big new update to ChatGPT.

Cofounder and CEO Sam Altman had teased "new stuff" coming to ChatGPT and GPT-4 , the AI model that powers its chatbot, and told his followers to tune in Monday at 1 p.m. ET for its "Spring Update" to learn more.

Also ahead of time, Altman ruled that the event would reveal GPT-5 or a new OpenAI search engine, which is reportedly in the works. OpenAI is reportedly planning to eventually take on internet search giant Google with its own AI-powered search product.

But the big news on Monday was OpenAI's new flagship AI model, GPT-4o, which will be free to all users and "can reason across audio, vision, and text in real time." It was CTO Mira Murati who delivered the updates with no appearance on the livestream from Altman.

There were a ton of demos intended to demonstrate the real-time smarts of GPT-4o.

OpenAI researchers showed how the new ChatGPT can quickly translate speech and help with basic linear algebra using its visual capabilities. The use of the tech on school assignments has been a polarizing topic in education since it first launched.

Say hello to GPT-4o, our new flagship model which can reason across audio, vision, and text in real time: https://t.co/MYHZB79UqN Text and image input rolling out today in API and ChatGPT with voice and video in the coming weeks. pic.twitter.com/uuthKZyzYx — OpenAI (@OpenAI) May 13, 2024

OpenAI posted another example to X of how one can interact with the new ChatGPT bot. It resembled a video call, and it got pretty meta.

In the video, ChatGPT takes in the room around it, discerns it's a recording setup, figures it might have something to do with OpenAI since the user is wearing a hoodie, and then gets told that the announcement has to do with the AI — it is the AI. It reacts with a voice that sounds more emotive.

OpenAI also announced the desktop version of ChatGPT, and a new and improved user interface.

In addition to GPT-4o and ChatGPT, OpenAI's other products include its AI-powered image generator DALL-E , its unreleased text-to-video generator Sora , and its GPT app store.

You can catch up on our liveblog of the event below.

That’s a wrap! OpenAI concludes the event without an appearance from Altman.

OpenAI says text and image input for GPT-4o-powered ChatGPT is launching today. Meanwhile, voice and video options will drop in the coming weeks, the company said.

Although Altman didn't step in front of the camera, the CEO posted videos from the audience on X.

He also teases "more stuff to share soon."

GPT-4o can also break down charts

The new AI model can interact with code bases, the OpenAI execs say. The next demo shows it analyzing a chart from some data.

It's a plot of global temperatures. GPT-4o gives some takeaways from what it sees, and CTO Mira Murati asks about the Y axis, which the AI explains.

ChatGPT reads human emotions — with a stumble

For the last live demo of the day, Zoph holds his phone up to his face and asks ChatGPT to tell him how he looks. Initially, it identifies him as a "wooden surface" — a reference to an earlier photo he had shared.

But after a second try, the model gives a better answer.

"It looks like you're feeling pretty happy and cheerful," ChatGPT says, noting the small smile on Zoph's face.

In one of the final tests, ChatGPT becomes a translator

In response to a request from an X user, Murati speaks to ChatGPT in Italian.

In turn, the bot translates her query into English for Zoph and Chen.

"Mike, she wonders if whales could talk, what would they tell us?" she said in English after hearing Murati's Italian.

It's pretty impressive.

The video demo shows how it could help with math homework, including basic linear algebra

OpenAI Research Lead Barret Zoph walks through an equation on a whiteboard (3x+1=4), and ChatGPT gives him hints as he finds the value of x — making it basically a real-time math tutor.

At the beginning, the bot jumped the gun.

"Whoops, I got too excited," it said after it tried to solve the math problem hadn't been uploaded yet.

But it then walked him through each step, recognizing his written work as he tried to solve the equation.

It was able to recognize math symbols, and even a heart.

OpenAI's first demo: Talking to GPT-4o

It's demo time!

The new bot has a voice that sounds like an American female, but no word yet if you can change it.

OpenAI Research Lead Mark Chen pulled out ChatGPT on his phone and asks for advice on giving a live presentation using Voice Mode.

"Mark, you're not a vacuum cleaner," it responds when he hyperventilates, appearing to perceive his nervousness. It then tells him to moderate his breathing.

Some big changes, you can interrupt the AI now, and there shouldn't be the usual 2 or 3-second delay with GPT-4o.

It can also detect emotion, according to OpenAI.

GPT-4o will have improved voice capabilities

Murati emphasizes the necessity of safety with the real-time voice and audio capabilities of the new GPT-4o model.

She says OpenAI is "continuing our iterative deployment to bring all the capabilities to you."

Murati says the big news is a "new flagship model" called GPT-4o.

The new model is called GPT-4o, and Murati says that OpenAI is making a "huge step forward" with ease of use with the new model.

It's free for users, and "allows us to bring GPT-4 class intelligence to our free users," Murati says.

And we're off!

The livestream began with CTO Mira Murati at OpenAI's offices.

OpenAI is going to be announcing 3 things today, she says. "That's it."

For those who want to watch live, you can view the whole event here.

OpenAI will be livestreaming its spring update, which kicks off in less than an hour.

Axel Springer, Business Insider's parent company, has a global deal to allow OpenAI to train its models on its media brands' reporting.

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