Calculating (1.4 x 106 m/s)2 - (3.5x104 m/s)2

[bumblebeeliz]

Homework Statement

(1.4 x 10⁶ m/s)² - (3.5x10⁴ m/s)²

Homework Equations

The Attempt at a Solution

We square them and then do we subtract them? Which would equal to:

(1.4 x 10⁶ m/s)² - (3.5x10⁴ m/s)²
(1.96 x 10¹² m²/s²) - (1.22x10⁸ m²/s²)
(1.95878 x 10⁴ m²/s² )

Is this correct?

 

[Doc Al]

We square them and then do we subtract them? Which would equal to:

(1.4 x 10⁶ m/s)² - (3.5x10⁴ m/s)²
(1.96 x 10¹² m²/s²) - (1.22x10⁸ m²/s²)

OK, except for a decimal mistake in one of your numbers.

(1.95878 x 10⁴ m²/s² )

No. How did you get this? (What's 10¹² - 10⁸? To subtract, rewrite the numbers so they have the same exponent.)

 

[bumblebeeliz]

I thought we could subtract the exponent. I am not sure how to re-write the numbers and make them have the same exponent.

 

[Doc Al]

I thought we could subtract the exponent.

Not when you are subtracting the numbers. Look what happens if you do:
10² - 10¹ ≠ 10¹
(100 - 10 = 90, not 10!)

I am not sure how to re-write the numbers and make them have the same exponent.

10² - 10¹ = 10x10¹ - 1x10¹ = (10-1)x10¹ = 90

1000 = 1x10³ = 10x10² = 100x10¹

 

[bumblebeeliz]

I see what you mean, but I am really not sure how to apply it in my situation. I basically factor it?

I also saw this thread: https://www.physicsforums.com/showthread.php?t=100697

But for some reason it is really not clicking how I am supposed to apply it.

I looked at some resources online and somebody suggested to divide them to allow the subtraction:

1.4 x 10¹² / 3.5 x 10⁸ = 0.4 x 10⁴

 

[Doc Al]

I see what you mean, but I am really not sure how to apply it in my situation. I basically factor it?

Not sure what you mean by "factoring". Here's the trick. If you reduce the exponent by 1, then you must increase the number by a factor of 10 (since this is base 10). Which just means move the decimal point to the right by 1.

1.96 x 10¹² = 19.6 x 10¹¹ = 196 x 10¹⁰ = 1960 x 10⁹

And so on...

I looked at some resources online and somebody suggested to divide them to allow the subtraction:

1.4 x 10¹² / 3.5 x 10⁸ = 0.4 x 10⁴

When you divide two numbers, then you can just subtract the exponents. But you're subtracting two numbers.

Note: I'm taking your first expression, where you subtract the two numbers, as given. So this is really a math question, not a physics question, since you haven't described the problem you're trying to solve. If you're not sure whether you should be subtracting or dividing, then describe the complete problem you're trying to solve.

 

[bumblebeeliz]

Ah! Its all about the tricks for beginners. Much better :)

= 19600 x 10⁸ - 1.22x10⁸
= 19598.7 x 10⁸

or

= 1.959878 x 10¹²

 

[bumblebeeliz]

One more question:

1.95978 x 10¹²/ 0.02

97.989 x 10¹² /2

48.969 x 10¹¹

4.9 x 10¹²

Is this correct? So when you divide, the exponent drops?

 

[Doc Al]

One more question:

1.95978 x 10¹²/ 0.02

97.989 x 10¹² /2

Careful. That "/2" shouldn't be there.

1.95978 x 10¹²/ 0.02 = (1.95978/ 0.02) x 10¹² = 97.989 x 10¹²

Note that the denominator didn't have an exponent:
0.02 = 0.02 x 10⁰

 

[bumblebeeliz]

Sorry, I meant this below: (scratch the first line)

97.989 x 10¹² /2

48.969 x 10¹¹

4.9 x 10¹²

(next time ill post the whole problem..sorry)

 

[Doc Al]

Sorry, I meant this below: (scratch the first line)

97.989 x 10¹² /2

48.969 x 10¹¹

No. The exponent doesn't change; just divide the numbers:
97.989 x 10¹² /2 = (97.989/2) x 10¹² = 48.969 x 10¹²

 

[bumblebeeliz]

Thanks for your patience :)