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### Course: Algebra 2 (Eureka Math/EngageNY) > Unit 3

Lesson 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

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# Approximating square roots to hundredths

Let's approximate the square root of 45 without a calculator. We'll explore how to find the perfect squares around 45 and use them to make an educated guess. Then, we'll refine our guess by squaring it to see how close we get to 45. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What are the differences between square roots and cube roots?(141 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.(9 votes)

- what is a perfect square and what does it mean?0:46(31 votes)
- A perfect square is a number that can be expressed as the product of two equal integers.

Examples of perfect squares:

* 9

o 9 is a perfect square becuase it can be expressed as 3 * 3 (the product of two equal integers)

* 16

o 16 is a perfect square becuase it can be expressed as 4 * 4 (the product of two equal integers)

* 25

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

* 5

o 5 is a not perfect square because it cannot be expressed as the product of two equal integers

* 7

o 7 is a not perfect square because you cannot express it as the product of two equal integers

hope this helps :)(47 votes)

- at1:26what does he mean by "nine of the way through it? Please help(17 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.(15 votes)

- I found this confusing a bit. Did anyone else?(12 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***(15 votes)

- why not just use Desmos calculator(4 votes)
- probably because if they allowed calculators people could just solve it with a built-in square root function and not have any actual knowledge of how it works (like what you seem to be suggesting), also, things like Desmos (and really, modern technology in general) only exist because people did math to find accurate and efficient ways to calculate things like this(8 votes)

- i still don't understand...(8 votes)
- Why not try watching the video again to see if it makes sense? It might take a while to fully understand, but you'll get there at some point :)(1 vote)

- How does he get 9/13(5 votes)
- The difference between 36 and 49 is 13, and 45 is 9 greater than 36, so it’s 9/13 of the way from 36 to 49(7 votes)

- how did he get the 9 in 9/13?(5 votes)
- Hello!

Do you understand the part where Sal says that 45 (and the square root of 45) is halfway between 49 (and the square root of 49) and 36 (and the square root of 36)?

At1:06, Sal explains that the difference between 49 and 45 is 4, and that the difference between 45 and 36 is 9. Therefore, the difference between 49 and 36 is 13.

He then puts the 9 above the 13 to get 9/13!

This was confusing for me too, but I hope this helped!(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!(10 votes)

## Video transcript

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.