Algebra 2 (Eureka Math/EngageNY)
Course: Algebra 2 (Eureka Math/EngageNY) > Unit 3Lesson 8: Topic C: Lesson 16: Rational and Irrational Numbers
- Approximating square roots
- Approximating square roots walk through
- Approximating square roots
- Comparing irrational numbers with radicals
- Comparing irrational numbers
- Approximating square roots to hundredths
- Comparing values with calculator
- Comparing irrational numbers with a calculator
- Proof: sum & product of two rationals is rational
- Proof: product of rational & irrational is irrational
- Proof: sum of rational & irrational is irrational
- Sums and products of irrational numbers
- Worked example: rational vs. irrational expressions
- Worked example: rational vs. irrational expressions (unknowns)
- Rational vs. irrational expressions
Approximating square roots to hundredths
Learn how to approximate the decimal value of √45 without using a calculator. Created by Sal Khan and Monterey Institute for Technology and Education.
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- What are the differences between square roots and cube roots?(132 votes)
- The cube root is the original number to the third power as an example, 3. 3 squared is 9. but nine times 3 or 3 cubed is 27. the cube root is 3 because 3*3*3=27.(5 votes)
- what is a perfect square and what does it mean?0:46(28 votes)
- A perfect square is a number that can be expressed as the product of two equal integers.
Examples of perfect squares:
o 9 is a perfect square becuase it can be expressed as 3 * 3 (the product of two equal integers)
o 16 is a perfect square becuase it can be expressed as 4 * 4 (the product of two equal integers)
o 25 is a perfect square becuase it can be expressed as 5 * 5 (the product of two equal integers)
NON examples of perfect squares:
… (more) * 8
o 8 is a not perfect square because you cannot express it as the product of two equal integers
o 5 is a not perfect square because it cannot be expressed as the product of two equal integers
o 7 is a not perfect square because you cannot express it as the product of two equal integers
hope this helps :)(39 votes)
- at1:26what does he mean by "nine of the way through it? Please help(15 votes)
- Take the distance from 36 to 49. that difference is 13 (49-36=13)
Next take the distance from 36 to 45. that difference 9.
So when you are going from 36 to 49, you will arrive at 45 at when you have completed nine 16ths of the trip.(11 votes)
- What is the difference between square roots and cube roots(3 votes)
- Squaring a number multiplies twice. Some squared numbers:
1² = 1 * 1
2² = 2 * 2
3² = 3 * 3
4² = 4 * 4
5² = 5 * 5
Cubing a number multiplies three times. Some cubed numbers:
1³ = 1 * 1 * 1
2³ = 2 * 2 * 2
3³ = 3 * 3 * 3
4³ = 4 * 4 * 4
5³ = 5 * 5 * 5
And so on.
But when we take the ROOT of a number, what we are actually doing is asking a question. When we get the square root of a number we are asking, "What number times what number equals the number we are squaring?" For example:
√4 = 2
The square root of 4 equals 2. Why? Because 2 times 2 equals 4. Another example:
√9 = 3
The square root of 9 equals 3. Why? Because 3 times 3 equals 9.
Now, the difference between square roots and cube roots is that with cube roots, we are asking a similar question, but the amount that the numbers need to multiply changes.
³√8 = 2
The cube root of 8 equals 2. Why? Because 2 times 2 times 2 equals 8. Another example:
³√27 = 3
The cube root of 27 equals 3. Why? Because 3 times 3 times 3 equals 27.
I hope you were able to understand and get through all that! It was a rather hefty manuscript. :)
Toodleoo! *tips hat*(11 votes)
- I do not think many of us are taking into account that finding square roots is just the opposite of finding the area of a square, And finding cube roots is just the opposite of finding the volume of a cube. You all may know this, but if you didn't, I hope you find this unit easier with your newfound knowledge!(7 votes)
- Wouldn't 6 + 9/13 be the square root of 45?(2 votes)
- This is an interesting question. It is true that 45 is 9/13 of the way from 36 to 49. However, because the square root function is a nonlinear function, the previous sentence does not mean that sqrt(45) is exactly 9/13 of the way from sqrt(36)=6 to sqrt(49)=7.
Since the graph of the square root function is concave down, sqrt(45) is larger than 6 + 9/13; sqrt(45) is about 6.708, but 6 + 9/13 is only about 6.692.
Visually, the graph of the function y=sqrt(x) on the interval 36<x<49 is above the graph of the line segment joining (36, 6) and (49, 7) but excluding the endpoints. This is why it makes sense for sqrt(45) to exceed 6 + 9/13. So 6 + 9/13 is only an approximation for sqrt(45).
Have a blessed, wonderful day!(9 votes)
- What does Sal mean when he says "This isn't a linear relationship" at1:21?(2 votes)
- Sal means that every time we increase the number inside the sqrt by a constant amount, the value of sqrt(x) doesn't increase by a proportional amount.
45 - 36 = 9.
sqrt(45) - sqrt(36) is about 0.7.
54 - 45 = 9 (again).
sqrt(54) - sqrt(45) is about 0.6.
If the relationship was linear, the difference between sqrt(54) and sqrt(45) would be 0.7 again, because in linear relationships the change in one variable is proportional to the change in another. However, the change in the value of sqrt(x) is not the same between x = 45 and x = 54(7 votes)
- What is a principal square root Sal mentions at0:20?(2 votes)
- The principle square root: Have you ever thought of it this way? Lets say you have √25. Now you would obviously automatically know that it is 5, right? But what about -5? You also know that -5 squared is 25 because a negative times a negative is a positive. Therefore, we call the positive square root (5) the principle square root to avoid confusion with the negative one. Hope this helped! :)(5 votes)
- Does anyone know how to square root with a power of 3(3 votes)
- To square root with a power of three, you will cube root the number.
For example, 4 to the power of three, or 4 squared is the cube root of 64, or 4*4*4.(3 votes)
- whats the differences to square roots and cube roots?(2 votes)
- A square root of a number returns the number which, multiplied by itself, gives the number. Cube root, on the other hand, returns the number which, multiplied by itself thrice, gives the number.(4 votes)
We are asked to approximate the principal root, or the positive square root of 45, to the hundredths place. And I'm assuming they don't want us to use a calculator. Because that would be too easy. So, let's see if we can approximate this just with our pen and paper right over here. So the square root of 45, or the principal root of 45. 45 is not a perfect square. It's definitely not a perfect square. Let's see, what are the perfect squares around it? We know that it is going to be less than-- the next perfect square above 45 is going to be 49 because that is 7 times 7-- so it's less than the square root of 49 and it's greater than the square root of 36. And so, the square root of 36, the principal root of 36 I should say, is 6. And the principal root of 49 is 7. So, this value right over here is going to be between 6 and 7. And if we look at it, it's only four away from 49. And it's nine away from 36. So, the different between 36 and 49 is 13. So, it's a total 13 gap between the 6 squared and 7 squared. And this is nine of the way through it. So, just as a kind of approximation maybe-- and it's not going to work out perfectly because we're squaring it, this isn't a linear relationship-- but it's going to be closer to 7 than it's going to be to 6. At least the 45 is 9/13 of the way. Let's see. It looks like that's about 2/3 of the way. So, let's try 6.7 as a guess just based on 0.7 is about 2/3. It looks like about the same. Actually, we could calculate this right here if we want. Actually, let's do that just for fun. So 9/13 as a decimal is going to be what? It's going to be 13 into 9. We're going to put some decimal places right over here. 13 doesn't go into 9 but 13 does go into 90. And it goes into 90-- let's see, does it go into it seven times-- it goes into it six times. So, 6 times 3 is 18. 6 times 1 is 6, plus 1 is 7. And then you subtract, you get 12. So, went into it almost exactly seven times. So, this value right here is almost a 0.7. And so if you say, how many times does 13 go into 120? It looks like it's like nine times? Yeah, it would go into it nine times. 9 times 3. Get rid of this. 9 times 3 is 27. 9 times 1 is 9, plus 2 is 11. You have a remainder of 3. It's about 0.69. So 6.7 would be a pretty good guess. This is 0.69 of the way between 36 and 49. So, let's go roughly 0.69 of the way between 6 and 7. So this is once again just to approximate. It's not necessarily going to give us the exact answer. We have to use that to make a good initial guess. And then see how it works. Let's try 6.7. And the really way to try it is to square 6.7. So 6.7 times-- maybe I'll write the multiplication symbol there-- 6.7 times 6.7. So, we have 7 times 7 is 49. 7 times 6 is 42, plus 4 is 46. Put a 0 now because we've moved a space to the left. So, now we have 6 times 7 is 42. Carry the 4. 6 times 6 is 36, plus 4 is 40. And so, 9 plus 0 is 9. 6 plus 2 is 8. 4 plus 0 is 4. And then we have a 4 right over here. And we have two total numbers behind the decimal point. One, two. So this gives us 44.89. So, 6.7 gets us pretty close. But we're still not probably right to the hundredth. Well, we're definitely not to the hundredths place. This since we've only gone to the tenths place right over here. So, if we want to get to 45, 6.7 squared is still less than 45, or 6.7 is still less than the square root of 45. So let's try 6.71. Let me do this in a new color. I'll do 6.71 in pink. So, let's try 6.71. Increase it a little bit. See if we go from 44.89 to 45. Because this is really close already. Let's just try it out. 6.71. So once again, we have to do some arithmetic by hand. We are assuming that they don't want us to use a calculator here. So, we have 1 times 1 is 1. 1 times 7 is 7. 1 times 6 is 6. Put a 0 here. 7 times 1 is 7. 7 times 7 is 49. 7 times 6 is 42, plus 4 is 46. And then we have two 0s here. 6 times 1 is 6. 6 times 7 is 42. Just have this new 4 here. 6 times 6 is 36, plus 4 is 40. Plus 40. It's interesting to think what we got incrementally by adding that one hundredth over there. Well, we'll see actually when we add all of this up. You get a 1. 7 plus 7 is 14. 1 plus 6 plus 9 is 16, plus 6 is 22. 2 plus 6 plus 2 is 10. And then 1 plus 4 is 5. And then we bring down the 4. And we have one, two, three, four numbers behind the decimal point. One, two, three, four. So, when you we squared 6.71. 6.71 squared is equal to 45.0241. So 6.71 is a little bit greater. So, let me make it clear now. We know that 6.7 is less than the square root of 45. And we know that is less than 6.71. Because when we square this, we get something a little bit over the square root of 45. But the key here is when we square this, so 6.7 squared got us 44.89 which is 0.11 away from 45. And then, if we look at 6.71 squared, we're only 2.4 hundredths above 45. So, this right here is closer to the square root of 45. So if we approximate to the hundredths place, definitely want to go with 6.71.