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### Using the Distributive Property

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Date: 03/06/2002 at 12:23:14
From: Melissa
Subject: Integers

This question is actually from my chemistry class:

-2.4222 x 10^-19 - (-6.0556 x 10^-20)

I tried typing it in as is on the calculator. As well I have tried

-2.18 x 10^-18 + .06556 x 10^-18

The teacher gave the answer as +1.81664 x 10^-19. I keep getting the
```

```
Date: 03/06/2002 at 12:44:45
From: Doctor Ian
Subject: Re: Integers

Hi Melissa,

The idea behind adding numbers like this is that if you have the same
exponent, you can use the distributive property:

2.13 * 10^-5 + 3.46 * 10^-5 = (2.13 + 3.46) * 10^-5

= 5.59 * 10^-5

It's the same as if you had a variable instead of an exponent:

2.13x + 3.46x = (2.13 + 3.46)x

= 5.59x

When you start with different exponents, you have to pick a common
exponent - usually one of the exponents you already have - and convert
both numbers to it:

2.13 * 10^-5 + 3.46 * 10^-6

= 21.3 * 10^-6 + 3.46 * 10^-6

= (21.3 + 3.46) * 10^-6

= 24.76 * 10^-6

Changing the exponent doesn't change the sign:

-2.13 * 10^-5 + 3.46 * 10^-6

= -21.3 * 10^-6 + 3.46 * 10^-6

= (-21.3 + 3.46) * 10^-6

= -(21.3 - 3.46) * 10^-6

= -17.84 * 10^-6

When you use the distributive property explicitly, as I have here, it
can help you avoid making sign errors.  (And you don't even have to
enter the exponents in the calculator.)

For your first problem, I get

-2.4222 * 10^-19 - (-6.0556 * 10^-20)

= -2.4222 * 10^-19 - (-0.60556 * 10^-19)

= (-2.4222 - -0.60556) * 10^-19

= (-2.4222 + 0.60556) * 10^-19

= -(2.4222 - 0.60556) * 10^-19

= 1.81664 * 10^-19

This is the answer your teacher got. Can you see now why it's correct?

Note that writing (or typing) numbers with lots of decimals is a good
way to make careless transcription errors. This is one of the reasons
that variables were invented. I would do the problem this way:

-2.4222 * 10^-19 - (-6.0556 * 10^-20)

= -a * 10^-19 - (-b * 10^-20)

= -a * 10^-19 - (-b/10 * 10^-19)

= (-a - -b/10) * 10^-19

= (-a + b/10) * 10^-19

= -(a - b/10) * 10^-19

= -(2.4222 - 0.60556) * 10^-19

= 1.81664 * 10^-19

This makes it much easier to check that you haven't messed up a sign,
and it's a lot easier on your fingers.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 03/06/2002 at 18:08:13
From: Melissa
Subject: Integers

Thanks for getting back to me. Still having difficulty. I understand
everything up until here:

= -(2.4222 - 0.60556) * 10^-19

= 1.81664 * 10^-19

Where did that negative sign in front of the brackets come from?
Please refresh my mind on this procedure as it has really been about
10 years since I learned all this the first time!
```

```
Date: 03/06/2002 at 18:37:33
From: Doctor Ian
Subject: Re: Integers

Hi Melissa,

That's a shortcut for a particular use of the distributive property,

a(b + c) = ab + ac

The longer version looks like this:

-2.422 + 0.60556

= (-1 * 2.422) + (-1 * -0.60556)    Each term is -1 times something.

= -1 * (2.422 + -0.60556)           Distributive property.

= -1 * (2.422 - 0.60556)            To add a negative, subtract
a positive.

= -(2.422 - 0.60556)                Keep the sign.

In practice, when you have something like

(a - b + c - d + e ...)

changing the sign of the whole thing also changes the signs of the
individual variables:

(a - b + c - d + e ...)

= -(-a + b - c + d - e ...)

Again, the distributive property explains why this works; but in
practice, you just use the shortcut.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents

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