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# 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.

## Want to join the conversation?

• What are the differences between square roots and cube roots? • what is a perfect square and what does it mean? •  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 :) • 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.
• What is the difference between square roots and cube roots • 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*
• 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! • Wouldn't 6 + 9/13 be the square root of 45? • 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!
• What does Sal mean when he says "This isn't a linear relationship" at ? • 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.
For example:
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
• What is a principal square root Sal mentions at ?   