Math Moms: How do you approximate large powers?

[bnrmom]

My son had a math problem that said: There are approximately 7 to the power of 10 people living in the U.S. About how many people is this?

 

I really didn't know how to approximate this, and certainly would want a calculator to solve it. Am I missing something?

 

And while we're at it, is there some trick to finding the square root of large numbers? For instance, the square root of 529, without a calculator?

 

Thank you!

[mirth]

You can memorize a few easy powers of ten and then go from there.

 

 

10^3= thousand

10^6= million

10^9= billion

10^12= trillion

 

So 10^7 would be 10x10^6, or ten million. 10^8 would be 100xmillion, or one hundred million.

[nmoira]

And while we're at it, is there some trick to finding the square root of large numbers? For instance, the square root of 529, without a calculator

How precise do you wish to be? There is a pencil-and-paper algorithm (not intuitive), but many use guess and check (intuitive) if you are only going to a decimal place or two.

[Perry]

My son had a math problem that said: There are approximately 7 to the power of 10 people living in the U.S. About how many people is this?

 

 

 

You can memorize a few easy powers of ten and then go from there.

 

 

10^3= thousand

10^6= million

10^9= billion

10^12= trillion

 

So 10^7 would be 10x10^6, or ten million. 10^8 would be 100xmillion, or one hundred million.

I think she is asking about 7^10, rather than 10^7. :)

 

Since 7^2 is 49, you could round that to 50.

 

7^10= (7^2)^5 or ~50^5

From there, it's fairly easy to break it down and do it mentally.

[mirth]

I think she is asking about 7^10, rather than 10^7. :)

 

Since 7^2 is 49, you could round that to 50.

 

7^10= (7^2)^5 or ~50^5

From there, it's fairly easy to break it down and do it mentally.

 

Oops! you're right. I was thinking 10^7 was way too low for the answer.

[Perry]

 

And while we're at it, is there some trick to finding the square root of large numbers? For instance, the square root of 529, without a calculator?

 

Thank you!

Yes, you can write the number as a product of prime numbers. If it is a perfect square, you will have an even number of each prime. For example

 

15876 = 2*2*3*3*3*3*7*7

 

Multiply one of each pair together (2*3*3*7)=126

Check: 126^2=15876

 

Obviously, you have to know how to prime factor to do this.

 

For your example, 529, this won't really work as it is already a product of primes. For this one, I'd just guess and check.

 

First, estimate:

20^2=400 too small

30^2=900 too big

25^2=625 too big

 

At this point, I know that if it is a perfect square it has to be 23, as that is the only number between 20 and 25 that multiplied by itself will leave a 9 in the ones column.

 

If it isn't a perfect square, I'd either round it to a more manageable number or guess and check.

[bnrmom]

I think she is asking about 7^10, rather than 10^7. :)

 

Since 7^2 is 49, you could round that to 50.

 

7^10= (7^2)^5 or ~50^5

From there, it's fairly easy to break it down and do it mentally.

Ah, that's the piece I was missing. I had forgotten about breaking the powers up into individual factors to make it easier.

 

Thank you everyone for the tips - I'm going to need it to get through 6th grade math!

[Bassoonaroo]

If you find these things interesting "How Many Licks" is a good read as is "Guesstimation" or for younger kids "Every Day in the World" or "Every Year in the World" (not sure of those titles...).

 

But with How Many Licks, be sure to edit it a bit because there are two quick chapters on how many folks are having tea and how many people are really really enjoying that tea right this moment.

 

Of course, I am neither having nor enjoying tea at this moment because I am typing...